Giáo trình giải tích lồi (tài liệu tham khảo)

Such applications are: the study of well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, best approximation prob- lems, characterizations of conve[r]

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Zalinescu, C ,

1952-Convex analysis in general vector spaces / C Zalinescu

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1 Convex functions 2 Convex sets 3 Functional analysis 4 Vector spaces

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To the memory of my parents

Casandra and Vasile Zalinescu

Preface

The text of this book has its origin in a course we delivered to students for Master Degree at the Faculty of Mathematics of the University "Al I Cuza" Ia§i, Romania

One can ask if another book on Convex Analysis is needed when there are many excellent books dedicated to this discipline like those written by R.T Rockafellar (1970), J Stoer and C Witzgall (1970), J.-B Hiriart-Urruty and C Lemarechal (1993), J.M Borwein and A Lewis (2000) for finite dimensional spaces and by P.-J Laurent (1972), I Ekeland and

R Temam (1974), R.T Rockafellar (1974), A.D Ioffe and V.M Tikhomirov (1974), V Barbu and Th Precupanu (1978, 1986), J.R Giles (1982), R.R Phelps (1989, 1993), D Aze (1997) for infinite dimensional spaces

We think that such a book is necessary for taking into consideration new results concerning the validity of the formulas for conjugates and sub-differentials of convex functions constructed from other convex functions

by operations which preserve convexity, results obtained in the last 10-15 years Also, there are classes of convex functions like uniformly convex, uniformly smooth, well behaving, well conditioned functions that are not studied in other books Characterizations of convex functions using other types of derivatives or subdifferentials than usual directional derivatives

or Fenchel subdifferential are quite recent and deserve being included in a book All these themes are treated in this book

We have chosen for studying convex functions the framework of locally convex spaces and the most general conditions met in the literature; even when restricted to normed vector spaces many results are stated in more general conditions than the corresponding ones in other books To make

vii

this possible, in the first chapter we introduce several interiority and ness conditions and state two strong open mapping theorems

closed-In the second chapter, besides the usual characterizations and properties

of convex functions we study new classes of such functions: convex, closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions, respectively; note that the classes of li-convex and lcs-closed functions have very good stability properties This will give the possibility

cs-to have a rich calculus for the conjugate and the subdifferential of convex functions under mild conditions In obtaining these results we use the method of perturbation functions introduced by R.T Rockafellar The main tool is the fundamental duality formula which is stated under very general conditions by using open mapping theorems

The framework of the third chapter is that of infinite dimensional normed vector spaces Besides some classical results in convex analysis we give characterizations of convex functions using abstract subdifferentials and study differentiability of convex functions Also, we introduce and study well-conditioned convex functions, uniformly convex and uniformly smooth convex functions and their applications to the study of the geometry of Banach spaces In connection with well-conditioned functions we study the sets of weak sharp minima, well-behaved convex functions and global error bounds for convex inequality systems The chapter ends with the study of monotone operators by using convex functions

Every chapter ends with exercises and bibliographical notes; there are more than 80 exercises The statements of the exercises are generally ex-tracted from auxiliary results in recent articles, but some of them are known results that deserve being included in a textbook, but which do not fit very well our aims The complete solutions of all exercises are given The book ends with an index of terms and a list of symbols and notations

Even if all the results with the exception of those in the first section are given with their complete proofs, for a successful reading of the book a good knowledge of topology and topological vector spaces is recommended Finally I would like to thank Prof J.-P Penot and Prof A Gopfert for reading the manuscript, for their remarks and encouragements

C Zalinescu March 1, 2002 Ia§i, Romania

Contents

Preface vii Introduction xi

Chapter 1 Preliminary Results on Functional Analysis 1

1.1 Preliminary notions and results 1

1.2 Closedness and interiority notions 9

1.3 Open mapping theorems 19

1.4 Variational principles 29

1.5 Exercises 34 1.6 Bibliographical notes 36

Chapter 2 Convex Analysis in Locally Convex Spaces 39

2.1 Convex functions 39

2.2 Semi-continuity of convex functions 60

2.3 Conjugate functions 75

2.4 The subdifferential of a convex function 79

2.5 The general problem of convex programming 99

2.6 Perturbed problems 106

2.7 The fundamental duality formula 113

2.8 Formulas for conjugates and e-subdifferentials, duality relations

and optimality conditions 121

2.9 Convex optimization with constraints 136

2.10 A minimax theorem 143

2.11 Exercises 146

2.12 Bibliographical notes 155

ix

Chapter 3 Some Results and Applications of Convex

Analy-sis in N o r m e d Spaces 159

3.1 Further fundamental results in convex analysis 159

3.2 Convexity and monotonicity of subdifferentials 169

3.3 Some classes of functions of a real variable and differentiability

of convex functions 188

3.4 Well conditioned functions 195

3.5 Uniformly convex and uniformly smooth convex functions 203

3.6 Uniformly convex and uniformly smooth convex functions on

bounded sets 221

3.7 Applications to the geometry of normed spaces 226

3.8 Applications to the best approximation problem 237

3.9 Characterizations of convexity in terms of smoothness 243

3.10 Weak sharp minima, well-behaved functions and global error

bounds for convex inequalities 248

Introduction

The primary aim of this book is to present the conjugate and subdifferential

calculus using the method of perturbation functions in order to obtain the

most general results in this field The secondary aim is to give important

applications of this calculus and of the properties of convex functions Such

applications are: the study of well-conditioned convex functions, uniformly

convex and uniformly smooth convex functions, best approximation

prob-lems, characterizations of convexity, the study of the sets of weak sharp

minima, well-behaved functions and the existence of global error bounds

for convex inequalities, as well as the study of monotone multifunctions by

using convex functions

The method of perturbation functions is based on the "fundamental

duality theorem" which says that under certain conditions one has

inf $ ( i , 0) = max_ ( - $*(0,y*)) (FDF)

For many problems in convex optimization one can associate a useful

perturbation function We give here four examples; see [Rockafellar (1974)]

for other interesting ones

Example 1 (Convex programming; see Section 2.9) Let f,9i, -,9n '•

X ->• E be proper convex functions with d o m / n C\7=i domgi ^ 0 The

problem of minimizing f(x) over the set of those x S X satisfying gi{x) < 0

for a l i i = 1 , , n is equivalent to the minimization of $(x, 0) for x 6 X,

where

- {

$ : I x F 4 l , *{x,y):={ f{x) * *(x) <yiV 1 < i <n,

1 +00 otherwise,

and Y := En; the element y* obtained from the right-hand side of (FDF)

will furnish the Lagrange multipliers

Example 2 (Control problems) Let F : X xY -±Rbe a, proper convex

function and A : X —> Y a linear operator A control problem (in its abstract form) is to minimize F(x,y) for x € X and y = Ax + yo- The perturbation function to be considered is $ : X x Y —> M defined by

$(x,y) := F{x,Ax + y 0 +y)

Example 3 (Semi-infinite programming) We are as in Example 1 but

{ 1 , , n } is replaced by a general nonempty set I\ In this case Y = W and $(x,y) := f(x) if gt(x) < yi for all i £ / , $(x,y) := co otherwise

Formula (FDF), or more precisely the Fenchel-Rockafellar duality mula, can also be used for deriving results similar to that in the next ex-ample

for-Example 4 ([Simons (1998b)]; see Exercise 2.37) Let X be a linear space,

(Y, ||-||) be a normed linear space, A : X —> Y be a linear operator, y 0 £ Y and / : X —> M be a proper convex function Then f(x) + \\Ax + yo\\ >

0 for all x € X if and only if there exists y* € Y* such that f(x) —

convexity (see Section 2.8)

The formula (FDF) is automatically valid when infx €x $(a;,0) = - c o

and is equivalent to the subdifferentiability at 0 £ Y of the marginal tion h : Y -> M, h(y) := mf xeX $(x,y), when i n fx ex $(#,0) £ E A

func-sufficient condition for this is the continuity of the restriction of h to the

affine hull of its domain at 0; note that 0 is in the relative algebraic interior

of the domain of h in this case (without this condition one can give simple examples in which the subdifferential of h at 0 is empty)

Considering the multifunction R : I x l = j 7 whose graph is the set

grTl = {(x,t,y) | (x,y,t) £ e p i $ } , the continuity of / ^ ( d o m ^ ) at 0 is

ensured if It is relatively open at some (a;0,io) with (a;o,io,0) € gr7? This fact was observed for the first time by Robinson (1976) This remark shows the importance of open mapping theorems for convex multifunctions in con-vex analysis In Banach spaces such a result is the well-known Robinson-

Introduction xm

Ursescu theorem The preceding examples show that the consideration of

more general spaces is natural: In Example 3 Y is a locally convex space while in Example 4 X can be endowed with the topology a(X, X') The

original result of Ursescu (1975) is stated in very general topological tor spaces The inconvenient of Ursescu's theorem is that one asks the multifunction to be closed, condition which is quite strong in certain situ-ations For example, when calculating the conjugate or subdifferential of

vec-max(/, g) with / , g proper lower semicontinuous convex functions one has

to evaluate conjugate or the subdifferential of 0 • / + 1 • g which is not lower

semicontinuous convex Fortunately we dispose of another open mapping theorem in which the closedness condition is replaced by a weaker one, but one must pay for this by asking (slightly) more on the spaces involved

As said above, the second aim of the book is to give some interesting applications of conjugate and subdifferential calculus, less treated in other books

In many algorithms for the minimization problem (P) min f(x), s.t x G

X, one obtains a sequence (x n ) which is minimizing (i.e (f(x n )) -» inf / )

or stationary (i.e (da/(z„)(0)) —> 0) It is important to know if such a sequence converges to a solution of (P) Assuming that S := a r g m i n / :=

{x | f(x) = i n f / } 7^ 0, one says that / is well-conditioned if (ds(x n )) -» 0

whenever (x n ) is a minimizing sequence, and / is well-behaved

(asymptot-ically) if (x n ) is minimizing whenever (x n ) is a stationary sequence; when

5 is a singleton conditioning reduces to the known notion of posedness in the sense of Tikhonov If / is well-conditioned with linear rate the set argmin / is a set of weak sharp minima When / is convex, we es-tablish several characterizations of well-conditioning using the conjugate or the subdifferential of / When 5 is a singleton one of the characterizations

well-is close to uniform convexity of / at a point

One says that the proper function / : (X, ||-||) ->• K is strongly convex if

f(\x + (1 - X)y) < Xf(x) + (1 - X)f(y) - f A(l - A) ||z - yf

for some c > 0 and for all x, y € d o m / , A G [0,1] This notion is not very

adequate for non-Hilbert spaces; for general normed spaces, one says that

/ is uniformly convex if there exists p : 1+ -+ E+ with p(0) = 0 such that

f(Xx + (1 - X)y) < Xf(x) + (1 - X)f(y) - A(l - X)p (\\x - y\\)

for all x,y e d o m / and all A € [0,1] From a numerical point of view

the class of uniformly convex functions is important because on a Banach space every uniformly convex and lower semicontinuous proper function has a unique minimum point and the corresponding minimization prob-lem is well-conditioned It turns out that uniformly convex functions have very nice characterizations using their conjugates and subdifferentials The dual notion for uniformly convex function is that of uniformly smooth con-vex function An important fact is that / is uniformly convex (uniformly smooth) if and only if /* is uniformly smooth (uniformly convex)

Another interesting application of convex analysis is in the study of monotone operators This became possible by using a convex function associated to a multifunction introduced by M Coodey and S Simons So, one obtains quite easily characterizations of maximal monotone operators, local boundedness of monotone operators and maximal monotonicity of the sum of two maximal monotone operators using continuity properties

of convex functions, the formula for the subdifFerential of a sum of convex functions and a minimax theorem (whose proof is also included)

A more detailed presentation of the book follows

The book is divided into three chapters, every chapter ending with ercises and bibliographical notes; there are more than 80 exercises It also includes the complete solutions of the exercises, the bibliography, the list

ex-of notations and the index ex-of terms

No prior knowledge of convex analysis is assumed, but basic knowledge

of topology, linear spaces, topological (locally convex) linear spaces and normed spaces is needed

In Chapter 1, as a preliminary, we introduce the notions and results of functional analysis we need in the rest of the book For easy reference, in Section 1.1 we recall several notions, notations and results (without proofs) which can be found in almost all books on functional analysis; let us men-tion four separation theorems for convex sets, the Dieudonne and Alaoglu-Bourbaki theorems, as well as the bipolar theorem

In Section 1.2 we introduce cs-closed, cs-complete, lcs-closed (i.e lower cs-closed) and ideally convex, bcs-complete, li-convex (i.e lower ideally con-

vex) sets and prove several results concerning them We point out the good stability properties of li-convex and lcs-closed sets We also introduce two

conditions denoted (Hx) and (Hwa;) which refer to sets in product spaces

that are stronger than the cs-closedness and ideal convexity, but weaker than the cs-completeness and bcs-completeness of the sets, respectively

Then, besides the classical algebraic interior A 1 and relative algebraic

in-Introduction xv

terior % A of a subset A of a linear space X, we introduce, when X is a

topological vector space, the sets lc A and tb A, which reduce to l A when the

affine hull aff A of A is closed or barreled, respectively, and are the empty

set otherwise The quasi interior of a set and united sets are also studied

In Section 1.3 we state and prove the famous Ursescu's theorem as well

as a slight amelioration of Simons' open mapping theorem As application

of these results one reobtain the Banach-Steinhaus theorem and the closed graph theorem as well as two results of 0 Carja which are useful in control-lability problems Because the notions (with the exception of cs-closed and ideally convex sets) and results from Sections 1.2 and 1.3 are not treated

in many books (to our knowledge only [Kusraev and Kutateladze (1995)] contains some similar material), we give complete proofs of the results The chapter ends with Section 1.4 in which we state and prove the Ekeland's variational principle, the smooth variational principle of Borwein and Preiss, as well as two (dual) results of Ursescu which generalize Baire's theorem

Chapter 2 is dedicated, mainly, to conjugate and e-subdifferential lus Because no prior knowledge of convex analysis is assumed, we introduce

calcu-in Section 2.1 convex functions, give several characterizations uscalcu-ing the graph, or the gradients in case of differentiability, point out the operations which preserve convexity and study the important class of convex functions

epi-of one variable; the existence epi-of the (e-)directional derivative and some epi-of its properties are also studied We close this section with a characterization

of convex functions using the upper Dini directional derivative

Section 2.2 is dedicated to the study of continuity properties of convex functions To the classes of sets introduced in Section 1.2 correspond cs-closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions We mention the fact that almost all operations which preserve convexity also preserve the lcs-closedness and the li-convexity of functions

as seen in Proposition 2.2.19 The most part of the results of this section are not present in other books; among them we mention the result on the convexity of a quasiconvex positively homogeneous function and the results on cs-closed, cs-complete, cs-convex, lcs-closed, ideally convex, bcs-complete and li-convex functions

Section 2.3 concerns conjugate functions; all the results are classical Section 2.4 is dedicated to the introduction and study of direct proper-ties of the subdifferential Using such properties one obtains easily the for-

mulas for the subdifferentials of Af and / i • • • • • / „ which are valid without

additional hypothesis The classical theorem which states that the differential of a proper convex function is nonempty and w*-compact at a continuity point of its domain, as well as the formula for the e-directional derivative as the support function of the e-subdifferential is also estab-lished The less classical result which states that the same formula holds for e > 0 when the function is not necessarily continuous (but is lower semi-continuous) is established, too We mention also Theorem 2.4.14 related

£-sub-to the subdifferential of sublinear functions; some of its statements are not very spread Other interesting results are introduced for completeness or further use

In Section 2.5 we introduce the general problem of convex ming and establish sufficient conditions for the existence and uniqueness

program-of solutions, respectively We mention especially Theorems 2.5.2 and 2.5.5; Theorem 2.5.2 ameliorates a result of Polyak (1966), which shows that the refiexivity of the space, needed in proving the existence of solutions, is al-most necessary, while Theorem 2.5.5 shows that the coercivity condition is essential for the existence of solutions

Section 2.6 is dedicated to perturbed functions One introduces primal and dual problems, the marginal function, and give some direct properties

of them Then one obtains the formula for the e-subdifferential of the marginal function using the (e + ^-subdifferentials (with 77 > 0) of the perturbed function Applying this result one obtains formulas for the e-subdifferential of several types of convex functions

In the main result of Section 2.7 we provide nine (non-independent) sufficient conditions which ensure the validity of the fundamental duality

formula (FDF) The most known of them is that (x 0 ,0) € d o m $ and

$(xo, •) is continuous at 0 for some x o £ l For the proof of the sufficiency

of some conditions one uses the open mapping theorems established in Section 1.3 A related result involves also a convex multifunction; this will be useful for obtaining the formulas for the conjugate and the e-sub-

differential of a function of the forms go A with A a densely defined and closed linear operator and of g o H with g being increasing and H convex

Section 2.8 is dedicated entirely to conjugate and e-subdifferential

cal-culus for convex functions The considered functions ip have the form:

ip(x) = F(x, A{x)) and tp — f + g o A with A a continuous linear

oper-ator, <p = / + g, ip(x) = ini{g(y) \ y £ C(x)} with 6 a convex process,

ip — g o H with H a convex operator and g an increasing convex

func-Introduction xvn

tion, cp = m a x { / i , , / „ } and tp = A 0 / 2 - Besides classical conditions

one points out very recent ones For the proof one constructs an adequate perturbation function and uses the fundamental duality theorem

In Section 2.9 we apply the fundamental duality theorem for ing necessary and sufficient optimality conditions in convex optimization problems with constraints These conditions involve the subdifferentials of the functions considered or the corresponding Lagrangian The results are well-known However we mention the formula for the normal cone to a level set stated in Corollary 2.9.5 for not necessarily finite-valued functions which is quite new

obtain-The minimax theorem presented in Section 2.10 will be used in the section dedicated to monotone multifunctions

Throughout Chapter 3 the involved spaces are normed spaces In tion 3.1 besides the classical theorems of Borwein, Br0ndsted-Rockafellar, Bishop-Phelps and Rockafellar (on the maximal monotonicity of the sub-differential of a convex function) we present a recent theorem of Simons and use it for a very short proof of Rockafellar's theorem (mentioned be-fore) As a consequence of the Br0ndsted-Rockafellar theorem we obtain other three conditions for the validity of the formulas for the conjugate and

Sec-subdifferential of the function F(-,A(-)) (and therefore for the functions

sev-In Section 3.3 we introduce the class A of functions tp : K+ —> R + with

y>(0) = 0 and several useful subclasses To any ip € A we associate tp# € A defined by <p&(t) = sup{ts — ip(s) | s > 0} These classes of functions turn

out to be useful in studying well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, as well as in the study

of the geometry of normed spaces As an illustration of the use of these classes of functions we study the differentiability of convex functions with

respect to arbitrary bornologies Using one of the characterizations and the Br0ndsted-Rockafellar theorem one obtains the following interesting result

of Asplund and Rockafellar: Let X be a Banach space and / : I - > l a

proper lower semicontinuous convex function; if / * is Frechet differentiable

at x* e int(dom/*) then V/*(x*) € X

In Section 3.4 we introduce the well-conditioned convex functions and give several characterizations of this notion using the conjugate and the subdifferential of the function An important special case is that of well-conditioning with linear rate This situation is studied in Section 3.10

In Section 3.5 we study uniformly convex and uniformly smooth convex functions, respectively To any convex function / one associates the gages

pj and 07 of uniform convexity and uniform smoothness, respectively The

gage pf has an important property: the mapping 0 < t i-» t~ 2 pf(t) is

nondecreasing Because a/* = ( p / ) * and a/ = ( p / * )# f°r a n v proper lower

semicontinuous convex function / , the mapping 0 < i 4 t~ 2 crf(t) is

non-increasing for such a function; moreover, one obtains that for such an / , /

is uniformly convex if and only if / * is uniformly smooth and / is uniformly smooth if and only if / * is uniformly convex Then one establishes many characterizations of uniformly convex functions and of uniformly smooth convex functions In these characterizations appear functions (gages or

moduli) belonging to different subclasses of A introduced in Section 3.3

These gages and moduli are sharp enough in order to obtain that / is

c-strongly convex if and only if /* is Frechet differentiable on X* and

V / * is c_1-Lipschitz Even if the results are established in general Banach spaces the natural framework for uniformly convex and uniformly smooth convex function is that of reflexive Banach spaces This is due to the fact that when there exists a proper lower semicontinuous and uniformly convex function on a Banach space whose domain has nonempty interior, the space

is necessarily reflexive

Section 3.6 is dedicated to the study of those convex functions which are uniformly convex on bounded sets and uniformly smooth on bounded sets, respectively Under strong coercivity of the function one shows that these notions are dual

In Section 3.7 we study the function /v : X ->• E, f^x) = / „ " tp(t) dt, where <p is a weight function, in connection with the geometric properties

of the norm So, one establishes characterizations of the strict

convex-ity, the smoothness and the reflexivity of X by the strict convexconvex-ity, the Gateaux differentiability of f v and the surjectivity of df v , respectively

Introduction xix

One obtains also characterizations of (local) uniform convexity and (local)

uniform smoothness of X with the help of the properties of f v For example

one obtains: X is uniformly convex •£> X* is uniformly smooth •& f v is

uni-formly convex on bounded sets -O (f v )* is uniformly smooth on bounded

sets •& {ftp)* is Frechet differentiable and V ( /v) * is uniformly continuous

on bounded sets Note that a part of the results of this section can be found

in the book [Cioranescu (1990)], but the proofs are different; note also that some notions are introduced differently in Cioranescu's book

Another application of convex analysis is emphasized in Section 3.8; here we apply the results on the existence, the uniqueness and the charac-terizations of optimal solutions of convex programs to the problem of the best approximation with elements of a convex subset of a normed space

In Section 3.9 it is shown that there exists a strong relationship between

the well-posedness of the minimization problem min/(:r) s.t x £ X, and

the differentiability at 0 of the conjugate / * of / ; when / is convex these properties are equivalent Using this result we establish a very interesting characterization of Chebyshev sets in Hilbert spaces and show that the class

of weakly closed Chebyshev sets coincides with the class of closed convex sets in Hilbert spaces

Section 3.10 deals with sets of weak sharp minima, well-behaved convex functions and the study of the existence of global error bounds for convex inequalities These notions were studied separately for a time, but they are intimately related As noted above, argmin / is a set of weak sharp minima for / exactly when / is well-conditioned with linear rate But the

inequality f{x) < 0 has a global error bound exactly when argmin[/]+

is a set of weak sharp minima for [/]+ := max(/, 0) We give several characterizations of the fact that argmin / is a set of weak sharp minima for / , one of them being the fact that up to a constant, the conjugate / *

is sublinear on a neighborhood of the origin Several numbers associated

to a convex function are introduced which are related to the conditioning number from numerical analysis Although the most part of the results from this section are stated in the literature in finite dimensional spaces,

we present them in infinite dimensions

The last section of this book, Section 3.11, is dedicated to the study of monotone multifunctions on Banach spaces We use in the presentation two recent articles of Simons The proofs are quite technical and use the lower

semicontinuous convex function XM associated to the multifunction M :

X =} X*, the minimax theorem and a few results of convex analysis One

obtains: two characterizations of maximal monotone multifunctions; the fact that the condition 0 6 int(domTi — domT2) is equivalent to other three

conditions involving dom Tj and dom XT { , and is sufficient for the maximal

monotonicity of T\ + T2; dom T and Im T are convex if X is reflexive and T

is maximal monotone; domT is convex if int(domT) ^ 0 and T is maximal monotone; T is locally bounded at XQ € (co(domX'))1 if T is a monotone

multifunction; Rockafellar's theorem on the local boundedness of maximal monotone multifunctions The result stating that for a maximal monotone

multifunction T on the Banach space X for which dom T is convex the local boundedness of T at x 6 dom T implies that x € int (dom T) seems to be

new When applied to the subdifferential of a proper lower semicontinuous

convex function / on the Banach space X, this result gives (for example): /

is continuous &• d o m / is open <& df is locally bounded at any x £ d o m /

The exercises are intended to exemplify the topics treated in the book Many exercises are auxiliary results spread in recent articles, although some

of them are extracted from other books Some exercises are important results which could be parts of textbooks, but which do not fit very well with the aim of the present book Among them we mention Exercise 3.11

on the Moreau regularization

Chapter 1

Preliminary Results on Functional

Analysis

1.1 Preliminary Notions and Results

In this section we introduce several notions and results on separation of sets

as well as some properties of topological vector spaces and locally convex spaces which are frequently used throughout the book, for easy reference

Let X be a real linear (vector) space Throughout this work we shall use the following notation (x, y being elements of X): [x, y] := {(1 — A)x + Xy \

A e [0,1]}, [x,y[:= {(1 - X)x + \y | A € [0,1[}, ]x,y[:= {(1 - X)x + Xy |

A 6]0,1[}, called closed, semi-closed and open segment, respectively Note that [x,x] =]x,x[= {x}\

If 0 ^ A, B C X, the Minkowski sum of A and B is A + B := {a +

b | a £ A,b £ B) Moreover, if x £ X, X £ R and 0 ^ T C R, then

x + A := A + x := A + {x}, A • A = {70 | A € A, a £ A} and XA — {A} • A

We shall consider that A + 0 = 0 and A • 0 = 0 • A = 0

A nonempty set A C X is star-shaped at a (G A) if [a, x] C A for all 2: £ A; A is convex if [x, y] C A for all x,y £ A; Ais a, cone if 1+ • A C A

(in particular 0 € A when A is a cone), R+ is the set of nonnegative reals;

A is afflne if Ax + (1 - X)y 6 A for all x,y e A and A e R; A is balanced if

Ax € ^4 for all x e A and A £ [-1,1]; A is symmetric if A = - A Hence ^4

is balanced if and only if A is symmetric and star-shaped at 0 We consider

that the empty set is convex and afflne It is easy to prove that

A is afflne o 3 a 6 l , 3X Q linear subspace of X : A = a + X 0

<=>Va€^4(3a€^4) : A — a is a linear subspace

When A is affine and a £ A, the linear space XQ := A — a is called the

l

linear space parallel to A; we consider that the dimension of A is

dimXo-Note that if (Aj)j€/ is a family of affine, convex, balanced subsets or

cones of X then f] ieI Ai has the same property (Exercise!); we use the

usual convention that HieO7^ = -^- Taking into account this remark, we can introduce the notions of affine, convex and conic hull of a set So, the

affine, convex and conic hull of the subset A of X are:

aff A := [){V C X \ A C V, V affine},

co A := P){C C X | A C C, C convex}, cone A := f]{C CX\AcC, C cone},

respectively Of course, the linear hull of the subset A of X is the linear

subspace spanned by A:

linA := P | { ^ o C X \ A C X 0 , X 0 linear subspace of X}

It is easy to verify (Exercise!) that

aff A = { V " XiXi n G N, (Ai)i<i < n C R, V " X, = 1,

(Xi)l<i<n C A> , coA= {Yl^-^i n S N, (Aj) C K+, (xt) C A, Yl"-^ = 1) '

cone A - {Ax | A > 0, x £ A} = 1+ • A,

where N is the set of positive integers

Let us mention some properties of the affine and convex hulls Consider

Y another linear space, T : X -> Y a linear operator, A,B c X, C C Y

nonempty sets Then: (i) aff(AxC) = aff Ax aff C; (ii) aff T(A) = T(aff A)

(iii) aft(A + B) = aff A + aff B; (iv) Vo € A : aff A = a + aff (A - A) (v) aff (A - A) = UA>oA(A - A) if A is convex; (vi) aff A = linA if 0 e A

(vii) co(A x C) = co A x coC; (viii) co T(A) = T(coA); (ix) co(A + 5 ) =

co A + coB; (x) co(cone A) = cone(co A) (Exercises!)

Let M C X be a linear subspace, and let A C X be nonempty; the

algebraic interior of A with respect to M is

a i n tM^ : = {a 6 X | Vrr € M, 36 > 0, VA € [0,6] : a + Xxe A}

It is clear that aint^f Ac A and that M C aff (A A) when aintM i ^ D

-Preliminary notions and results 3

We distinguish two important cases: (i) M = X; in this case we write

A % instead of aintM A; A 1 is called the algebraic interior of A, (ii) M =

aS(A — A); in this case aintM A is denoted by M and is called the relative

algebraic interior of A Therefore a £ A 1 ii and only if aff A = X and

a G 14 (Exercise!)

When the set A is convex we have (Exercise!) that:

V a G i : \m(A-a) = cone(A-A), whence aff A = a + cone(A — A) for every a £ A (hence cone(A — A) is the

linear subspace parallel to aff A),

Some properties of the algebraic interior are listed below Let 0 ^

A,B CX, i s l a n d A G E \ {0}; then: (i) *(x + A) =x + i A; (ii) ^XA) =

X • % (iii) A + B i C (A + B ) ' ; (iv) A + B l = {A + BY if B i = B;

(v) i A + i B c 1 {A + B); (vi) *(A + B) = A + *B if A, B are convex, VI / 0

and *B T^ 0; (vii) M 7^ 0 if A is convex and dim A < 00; (viii) if A is convex

then [a, x[ C VI for all a G M and i £ A

In the sequel the results will be established for real topological vector

spaces (tvs for short) or real locally convex spaces (lcs for short) When X

is a tvs it is well-known that the class Nx of closed and balanced

neigh-borhoods of 0 G X is a base of neighneigh-borhoods of 0; when X is a lcs then

the class J^ cx of the closed, convex and balanced neighborhoods of 0 G X is

also a base of neighborhoods of 0

If X, Y are real linear spaces, we denote by L(X, Y) the real linear space

of linear operators from X into Y The space L(X,R) is denoted by X'

and is called the algebraic dual of X; an element of X' is called a linear

functional When X, Y are topological vector spaces, we denote by &{X, Y)

the linear space of continuous linear operators from X into Y; the space

£ ( X , E) is denoted by X* and is called the topological dual of X

Let now A be an absorbing subset of the linear space X, i.e 0 £ A 1 ;

the Minkowski gauge of A is defined by

VxGX : pAns(a;) = max {PA (a;), pB(a;)},

Va; 6 X, 2/ € Y : PAxc(x,y) = max{p A

(x),pc(y)}-Other useful properties of the Minkowski gauge are mentioned in the

next result Recall that p : X -» M is sublinear if p(0) = 0, p(x + y) <

p(x) +p(y) [with the convention (+oo) + (—oo) = +oo] and p(Xx) = Xp(x)

for all x, y £ X, X € P := ]0, oo[; p is a semi-norm if p is a finite, sublinear

and even [i.e p(—x) = p(x) for every x € X] function

Proposition 1.1.1 Let A be a convex and absorbing subset of the linear

space X

(i) Then PA is finite, sublinear and A 1 = {x E X \ PA{X) < 1}; furthermore, if A is symmetric then PA is a semi-norm, too

(ii) Assume, moreover, that X is a topological vector space and V is a

neighborhood of Q £ X Then pv is continuous and

intV = {xeX\p v {x) < 1}, clV = {xeX \pv{x) < 1}

The following result will be useful, too

Theorem 1.1.2 Let C be a convex subset of the topological vector space

X Then

(i) c l C is convex;

(ii) if a £ int C and x £ cl C, then [a, x[ C int C;

(iii) i n t C is convex;

(iv) if int C ^ 0 then cl(int C) = cl C and int(cl C) = int C;

(v) if int C ^ 0 then C l = int C

Preliminary notions and results 5

Using the Minkowski gauge one obtains the geometrical versions of the

Hahn-Banach theorem, i.e separation theorems In the sequel we give

several separation theorems for convex subsets of topological vector spaces

or locally convex space

Theorem 1.1.3 (Eidelheit) Let A and B be two nonempty convex subsets

of the topological vector space X If int A ^ 0 and B flint A = 0 then there

exist x* e X* \ {0} and a £ E such that

VxeA,\/yeB : (x,x*) < a < (y,x*), (1.2)

or equivalently, supa;*(yl) < inf x*(B)

The separation condition (1.2) can be given in a different manner Let

x* e X* \ {0} and a £ t Consider the sets

H^ a :={x£X\(x,x*)<a}, H^, a ~{xeX\ (x,x*) <a},

H x *,a := {x e X \ (x,x*) = a } ;

similarly one defines H^ and H> a All these sets are convex and

non-empty The set H x *^ a is called a closed hyperplane, H<. a and H>* a are

called o p e n half-spaces, while H-. a and H~ a are called closed

half-spaces H x - <a , H-, a and H-, a are closed sets, while H<» a and H>, a are

open sets; moreover, c\H< a = H% a and {Hf, a ) 1 = int H~ >a = H< a

(Exercises!)

Theorem 1.1.3 states the existence of x* G X* \ {0} and a G K such that

A c H-, a and B C H-, a ; in this situation we say that H x * tC[ separates

A and B; the separation is proper when AUB <£ H x * i<x and the separation

is strict when A n H x *^ a = 0 or B n H x * ia = 0

When x 0 6 A and ffx*,a separates A and {xo} we say that ifx*,a is a

supporting hyperplane of A at xo; XQ is called a support point and

x* is called a support functional Therefore x* € X* \ {0} is a support

functional for A if and only if x* attains its supremum on A Generally,

H x * <a , with x* ^ 0, is a supporting hyperplane for A if A C B.-, a (or

i 4 c H ^i a) a n d A n i fs ,a^ 0

Corollary 1.1.4 Let A be a convex subset of the topological vector space

X having nonempty interior and x € A \ int A Then x is a support point

of A

In the case of locally convex spaces one has the following result for the

separation of two sets

T h e o r e m 1.1.5 Let X be a locally convex space and A,B C X be two

nonempty convex sets If A is closed, B is compact and An B = 0, then

there exist x* G X* \ {0} and ct\,a.2 G K such that

Vx G A, Vy e B : (x,x*) < a x < a 2 < (y,x*),

or equivalently, sup x* (A) < inf x* (B)

The two preceding results can be stated in a more general setting

T h e o r e m 1.1.6 Let A and B be two nonempty convex subsets of the

topological vector space X such that int(A — B) ^ 0 Then

0 £ i n t ( A - £ ) < £ • 3x* G X * \ { 0 } : supa;*(yl) < inf x*{B)

T h e o r e m 1.1.7 Let A and B be two nonempty convex subsets of the

locally convex space X Then

Q$d{A-B)&3x* eX* : supx*{A) < inf x*(B)

The preceding theorem shows the usefulness of having criteria for the

closedness of the difference (or sum) of two convex sets In order to give

such a criterion, let A be a nonempty convex subset of the topological vector

space X The recession cone of A is defined by

vecA := {u G X | Vo G A : a + uGA}

It is easy to show that rec A is a convex cone and A + rec A = A When A

is a closed convex set we have that

for every a 6 A In this case it is obvious that rec A is a closed convex

cone which is also denoted by Aoo It is easy to see that when X is a finite

dimensional separated topological vector space and A is a closed convex

nonempty subset of X, A^ = {0} if, and only if, A is bounded This is no

longer true when d i m X ~ oo

Example 1.1.1 Let X := £ p with p £ [l,oo] and A := {(x„)„>i G £ p |

\x n \ < n Vn G N} It is obvious that A is a closed convex set which is

not bounded because ne n G A for every n G N, but A^ = {0} Indeed, if

Preliminary notions and results 7

u = (u n ) G Aoo then tu G A for every t > 0; so \tu n \ < n for every t > 0,

whence u„ = 0 Hence u = 0

The following famous theorem was obtained in [Dieudonne (1966)]

Theorem 1.1.8 (Dieudonne) Let A, B be nonempty closed convex subsets

of the locally convex space X If A or B is locally compact and A^ n J3oo

is a linear subspace, then A — B is closed

In convex analysis (as well as in functional analysis) one often uses the

following sets associated to a nonempty subset A of the locally convex space

(x,x*)>-l}, (x,x*)>0}, (x,x*) = 0},

called the polar, the dual cone and the orthogonal space of A,

respec-tively One verifies easily that A° is a w*-closed convex set which contains

0, that A + is a w*-closed convex cone, and, finally, that A 1 - is a u;*-closed

linear subspace of X*, where w* = a(X*,X) is the weak* topology on X* Similarly, for 0 ^ B C X* we define the polar, the dual cone and the orthogonal space; for example, the polar of B is

B° := {x G X | Vx* G B : (x,x*) > - 1 }

It is obvious that B° is a closed convex set containing 0, B + is a closed

convex cone, and, finally, B 1 - is a closed linear subspace

One verifies easily that when A,BcX and A G P we have: (i) A° is convex and 0 G A°; (ii) A U {0} C (A°)° =: A°°; (hi) AcB => A° D B°; (iv) (AUB)° = A°nB°; (v) if 0 € AnB then (A + B)+ = (AUB)+ =A+n

B+; (vi) (\A)° = {A°; (vii) A° = A+ if A is a cone, and A° = A+ = A 1 - if

A is a linear subspace; (viii) (T(A))° = ( T * ) "1^0) , if T G &(X,Y), where

Y is another locally convex space

A very useful result is the bipolar's theorem Let X be a topological

vector space and A C X; the set coA := cl(coA) is called the closed

convex hull of the set A; it is the smallest closed convex set containing A

Similarly, coneA :— cl(coneyl) is called the closed conic hull of A

Theorem 1.1.9 (bipolar) Let A be a nonempty subset of the locally

con-vex space X Then

A 00 = co(A U {0}), A+ + = cone(co A), A ±JL = cl(lin A)

It follows that for the nonempty subset A of the lcs X one has: (a) A°° =

A o A is closed, convex and 0 € A; (b) A+ + = A <£> A is a closed convex cone; (c) A-"-1- = A <=> A is a closed linear subspace

Another famous result is the following

Theorem 1.1.10 (Alaoglu-Bourbaki) Let X be a locally convex space

and U C X be a neighborhood of the origin Then U° is w*-compact

We finish this preliminary section with some notions and results cerning completeness and metrizability of topological vector spaces

con-The subset A of the topological vector space X is complete

(quasi-complete) if every (bounded) Cauchy net (xi)i^i C A is convergent to an

element x £ A Of course, any complete set is closed and any closed subset

of a complete set is complete (Exercise!) Recall that the topological space

(X, T) is first countable if every element of X has a (at most) countable

base of neighborhoods Note that a subset A of a first countable tvs X

is complete if and only if every Cauchy sequence of A is convergent to an element of A; in particular, a first countable tvs is complete if and only if

it is quasi-complete (Exercise!)

We shall use several times the hypothesis that a certain topological vector space is first countable The next result refers to the first countability

of locally convex spaces

Proposition 1.1.11 Let (X,T) be a locally convex space Then

(i) (X, T) is first countable O 3 7 a (at most) countable family of

semi-norms on X such that r = rg> O r is semi-metrizable, i.e there exists a semi-metric d on X such that T = T^; the semi-metric d may be chosen to

be invariant to translations (i.e d(x + z,y + z) = d(x,y) for allx,y,z € X)

(ii) (X, T) is separated and first countable if and only if T is metrizable,

i.e there exists a metric d on X such that r = T&

Note that when the topology r of a locally convex space X coincides with the topology T<J determined by a semi-metric d invariant to translations, the completeness with respect to r and d are equivalent One says that the

locally convex space X is a Frechet space if X is complete and metrizable

It is obvious that every closed linear subspace of a Frechet space is Frechet,

Closedness and inferiority notions 9

too One says that the topological vector space X is barreled if every

absorbing, convex and closed subset of X is a neighborhood of 0 £ X As

application of the Baire theorem one obtains that every Frechet space is barreled

It is well-known that in a finite dimensional separated topological vector space any convex and absorbing set is a neighborhood of the origin

1.2 Closedness and Interiority Notions

Consider X a real topological vector space We say that the series Yl n >i Xn

is convergent (resp Cauchy) if the sequence (£n)n£N is convergent (resp

Cauchy), where S n := Y^k=i x k f °r every n £ N; of course, any convergent

series is Cauchy

Let A C X; by a convex series with elements of A we mean a series

of the form Yl m >i ^mX m with (Am) C IR+, {x m ) C A and Y, m >i ^™ = ^

if, furthermore, the sequence (x m ) is bounded we speak about a b-convex series We say that A is cs-closed if any convergent convex series with

elements of A has its sum in A;* A is cs-complete if any Cauchy convex

series with elements of A is convergent and its sum is in A Similarly, the set

A is called ideally convex if any convergent b-convex series with elements

of A has its sum in A and A is bcs-complete if any Cauchy b-convex series

with elements of A is convergent and its sum is in A It is obvious that any

closed set is ideally convex, every ideally convex set is convex, every

cs-complete set is cs-closed and every cs-complete convex set is cs-cs-complete; if X

is complete, then A C X is complete (bcomplete) if and only if A is closed (ideally convex) If Xo is a linear subspace of X and A is a cs-closed (ideally convex) subset of X, then Xo C\ A is a cs-closed (ideally convex) subset of Xo (endowed with the induced topology) Moreover, if A C X and B C Y are nonempty, then A x B is cs-closed (cs-complete, ideally convex, bcs-complete) if and only if A and B are cs-closed (cs-complete, ideally convex, bcs-complete) If X is first countable, a linear subspace

cs-XQ of X is cs-closed (cs-complete) if and only if it is closed (complete);

moreover, if X is a locally convex space, X 0 is closed if and only if Xo

is ideally convex Note also that T(A) is cs-closed (cs-complete, ideally convex, bcs-complete) if A C X is cs-closed (cs-complete, ideally convex,

'Because X may not be separated, in fact we ask that every limit of (S ) is in A

bcs-complete) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!), Y being another tvs We consider that the empty set is

convex, ideally convex, bcs-complete, cs-complete and cs-closed

It is worth to point out that when X is a locally convex space, every b-convex series with elements of X is Cauchy (Exercise!)

The class of cs-closed sets (and consequently that of ideally convex sets)

is larger than the class of closed convex sets, as the next result shows

Proposition 1.2.1 Let A C X be a nonempty convex set

(i) / / A is closed or open then A is cs-closed

(ii) If X is separated and dim A < oo then A is cs-closed

Proof, (i) Let £n >! Anin be a convergent convex series with elements

of A; denote by x its sum

Suppose that A is closed and fix a £ Ạ Then, for every n G N we have that X)fc=i ^kXk + (l — Yl'kLn+i ^*) a S Ạ Taking the limit for n -> oo,

we obtain that x G cl A = Ạ

Suppose now that A is open Assume that x $ Ạ By Theorem 1.1.3, there exists x* G X* such that (a - x, x") > 0 for every a G Ạ In particular

(x n — x,x*) > 0 for every n G N Multiplying by \ n > 0 and ađing for

n G N we get (since A„ > 0 for some n) the contradiction

0 < ^n > 1 A„ (xn - x, x*) = (X]„>i XnXn' x*)~ \52n>i A n) ^'x") = °'

Therefore x G Ạ So in both cases A is cs-closed

(ii) We prove the statement by mathematical induction on n := dim Ạ

If n = 0 A reduces to a point; it is obvious that A is cs-closed in this casẹ Suppose that the statement is true if dim A < n G N U {0} and show it for dim A = n + 1 Without any loss of generality we suppose that 0 G A; then Xo := aff A is a linear subspace with d i m X0 = n + 1

Because on a finite dimensional linear space there exists a unique separated linear topology and in such spaces the interior and the algebraic interior coincide for convex sets, we have that i A — intx0 A ^ 0 Let J2 n >i ^n%n

be a convergent convex series with elements of A and sum x Assume that

x fi Ạ Because A is convex, the set P :— {n G N | A„ > 0} is infinite,

and so we may assume that P — N Applying now Theorem 1.1.3 in XQ, there exists XQ G XQ \ {0} such that (x — x, XQ) > 0 for every x G Ạ But X)n>i An (x n —X,XQ) = 0 Since {x n —X,XQ) > 0 and X n > 0 for every

n, we obtain that (xn) C AQ := Ẵ)H x * t \, where A :— (X,XQ). Since

Closedness and inferiority notions 11

dimAo < dimH x * t \ = n, from the induction hypothesis we obtain the

contradiction x e A 0 c A Therefore x € A The proof is complete •

Other properties of cs-closed and ideally convex sets are given in the

following result

Proposition 1.2.2 (i) If Ai C X is cs-closed (resp ideally convex) for

every i £ I then P |i e / A{ is cs-closed (resp ideally convex)

(ii) / / Xi is a topological vector space and Ai C Xi is cs-closed (resp

ideally convex) for every i 6 I, then Ylizi -^-i is cs-closed (resp ideally

convex) in Yliei Xi (which is endowed with the product topology)

Proof The proof of (i) is immediate, while for (ii) one must take into

account that a sequence (xn)n €N C X := Yl ieI Xi converges t o x G X (resp

is bounded) if and only if (x ln ) converges to x l in Xi (resp is bounded) for

every i £ I •

We say that the subset C of Y is lower cs-closed (Ics-closed for short)

if there exist a Frechet space X and a cs-closed subset B of X x Y such

that C = Pry (B) Similarly, the subset C of Y is lower ideally convex

(li-convex for short) if there exist a Frechet space X and an ideally convex

subset B of X xY such that C = Pry (B) It is obvious that any cs-closed

(resp ideally convex) set is Ics-closed (resp li-convex), any Ics-closed set

is li-convex and any li-convex set is convex, but the converse implications

are not true, generally Note also that T(A) is Ics-closed (resp li-convex) if

A C X is Ics-closed (resp li-convex) and T : X -> Y is an isomorphism of

topological vector spaces (Exercise!) The classes of Ics-closed and li-convex

sets have very good stability properties as the following results show We

give only the proofs for the "li-convex" case, that for the "Ics-closed" case

being similar

Proposition 1.2.3 Suppose that Y is a Frechet space and C C F x Z is

a li-convex (Ics-closed) set Then Prz(C) is a li-convex (Ics-closed) set

Proof By hypothesis, there exists a Frechet space X and an ideally

convex subset B C X x (Y x Z) such that C = P r yxz ( B ) - Since I x 7 i s

a Frechet space and Prz(C) = P r z ( B ) , we have that Prz(C) is a li-convex

subset of Z •

Proposition 1.2.4 Let I be an at most countable nonempty set

(i) If Ci C Y is li-convex (Ics-closed) for every i £ I then (~) ieI Ci is

li-convex (Ics-closed)

(ii) / / Y{ is a topological vector space and Ci C Yi is li-convex

(Ics-closed) for every i 6 I, then Yli&iCi 8S li-convex (Ics-closed) in W^jYi

Proof, (i) For each i £ I there exist X» a Frechet space and an ideally

convex set B t C X» xY such that d = Pr Y (Bi) The space X := fliei -%i

is a Frechet space as the product of an at most countable family of Frechet

spaces Let

Bi •= {{(xj)jei,y) e X x Y | (xi,y) £ Bi}

Then Bi is an ideally convex set by Proposition 1.2.2(h) It follows that

B := f] ieI Bi is ideally convex by Proposition 1.2.2(i) Since Pr Y (B) =

Hie/ Ci, ^ f °u o w s that flie/ Ci is li-convex

(ii) For each i £ I there exist Xj a Frechet space and an ideally convex

set Bi cXiX Yi such that d = Pry; (Bi) By Proposition 1.2.2(h), ]J ieI B t

is an ideally convex subset of r i i e / ( ^ j x ^ ) - The space X := riig/ ^i 1S a

Frechet space; let Y := FJie/ ^i- Consider the set

B:= {((xi) ieI ,(yi)i eI ) £XxY\ (xi, yi ) £ Bi Vt 6 / }

Since T : I l i e / (X* x y4) - • X x y , T ( ( xi, yi)i 6/ ) := ((xi) ieI ,(yi) ie i) is

an isomorphism of topological vector spaces, B — T (Y\ ieI Bi) is ideally

convex As C := riig/ C* = ^ V Y(B), C is li-convex D

Before stating other properties of li-convex and lcs-closed sets, let us

define some notions and notations related to multifunctions

Let E,F be two nonempty sets; a mapping ft : E —• 2 F is called a

multifunction, and it will be denoted by ft : E =X F The set domft :=

{x £ E \ 5l(x) -£ 0} is called the d o m a i n of the multifunction 51; the i m a g e

of 51 is Imft := UigB^(a ;)i the g r a p h of 51 is the set grft :— {(x,y) \ y £

5l(x)} C E x F; the inverse of the multifunction 51 is the multifunction

ft"1 : F =} E defined by ft_1(?/) := {x £ E | y £ %(x)} Therefore

d o m f t -1 = Imft, I m f t -1 = domft and g r f t -1 = {(y,x) | (x,y) £ grft}

Frequently we shall identify a multifunction with its graph For A C E and

B C F one defines 51(A) := \J xeA 5l(x) and ft"1^) := Uj,eB#- 1(2/); i n

particular Imft = ft(.E) and domft = ft_1(F) If 8 : F =4 G is another

multifunction, then the composition of the multifunctions S and 51 is the

multifunction Soft :E=lG, (Soft) (a) := \J y€Ci{x) S(y) If ft, S : E =} F and

F is a linear space, the sum of ft and S is the multifunction ft + S : E =t F,

(ft + S)(x) :=5i(x) + S(x)

Closedness and interiority notions 13

Let now K : X r{ 7 ; we say that 51 is convex (closed, ideally

con-vex, bcs-complete, cs-concon-vex, cs-complete, li-concon-vex, lcs-closed)

if its graph is a convex (closed, ideally convex, bcs-complete, cs-convex,

cs-complete, li-convex, lcs-closed) subset of X x Y Note that 31 is convex

if and only if

V i , a ' eX, V A e [0,1] : \3l(x) + (l-\)3l(x')c3l(\x + {l-\)x')

Proposition 1.2.5 Let A, B C X, 31, S : X =4 Y and7:Y =} Z

(i) If X is a Frechet space and A, 31 are li-convex (resp lcs-closed),

then 01(A) is li-convex (resp lcs-closed)

(ii) If X is a Frechet space and A, B are li-convex (resp lcs-closed),

then A + B is li-convex (resp lcs-closed)

(hi) If Y is a Frechet space and 31, T are li-convex (resp lcs-closed),

then T o 31 is li-convex (resp lcs-closed)

(iv) IfY is a Frechet space and 31,$ are li-convex (resp lcs-closed), then

31 + § is li-convex (resp lcs-closed)

Proof, (i) We have that

%{A) =Pr Y ((AxY)ngrJl)

Using successively Propositions 1.2.4(h), 1.2.4(i) and 1.2.3, it follows that

31(A) is li-convex

(ii) Let T : X xX —> X, T(x,y) := x + y Since T is a continuous linear

operator, g r T is a closed linear subspace; in particular T is a li-convex

multifunction Since Ax B is li-convex, by (i) A + B = T(A x B) is a

li-convex set

(hi) We have that

gr(To3?) = P rXx 4 ( g r : R x Z) n (X x grT))

The conclusion follows from Propositions 1.2.4(h), 1.2.4(i) and 1.2.3

(iv) The sets

T~{(x,z,y,y')\x€X, y,y' eY, z = y + y'} C (X x Y) x (Y x Y),

A:= {(x,z,y,y') \ (x,y) G g r # , y',z € Y} and B := {(x,z,y,y') \ (x,y') G

grS, y,z € Y} are li-convex sets (the first being a closed linear subspace)

Let Y be another topological vector space and A C X xY; we introduce

the conditions (Ha;) and (Hwa;) below, where x refers to the component

xeX:

(Ha:) If the sequences ((x n , y n )) C A and (A„)„>! C ffi+ are such that

E „ > i K = 1, En> i ^nVn^as sum y and X)„>i A«a;n is Cauchy,

then the series X^n>i ^n x n * s convergent and its sum x G X verifies

{x,y)EA

(Hwi) If the sequences ({x n ,y n )) n>1 C A and (An)n>i C K+ are such

that ((x n ,y n )) is bounded, En>x An = 1, £ „ > i A„2/n has sum y

and ^r a > 1 Anxn is Cauchy, then the series X^n > 1 ^«x« ^s convergent

and its sum x € X verifies (x, y) 6 A

Of course, when X is a locally convex space, deleting "^2 n>1 Ana;n is

Cauchy" in condition (Hwa;) one obtains an equivalent statement

In the next result we mention the relationships among conditions (Ha;),

(Hwi), ideal convexity, cs-closedness, cs-completeness and convexity The

proof being very easy we omit it

Proposition 1.2.6 Let A C X xY and BcXxYxZbe nonempty

sets

(i) Assume that Y is complete Then B satisfies (H.(x,y)) if and only

if B satisfies (Ha;); B satisfies (Hwa;) if and only if B satisfies (Hw(x,j/))

(ii) Assume that X is complete Then A satisfies (Ha;) if and only if A

is cs-closed; A satisfies (Hwa;) if and only if A is ideally convex

(iii) Assume that Y is complete Then A satisfies (Ha;) if and only if A

is cs-complete; A satisfies (Hwa;) if and only if A is bcs-complete

(iv) If A satisfies (Ha;) then A satisfies (Hwx); if A satisfies (Hwa;)

then A is convex

(v) Assume that X is a locally convex space and Prx(A) is bounded If

A satisfies (Ha;) then Pry(^4) is cs-closed; if A satisfies (Hwa;) then Pry (A)

is ideally convex

We define now several interiority notions Let 0 / A c X We denote

by rint A the interior of A with respect to aff A, i.e rint A := intafj A A Of

Closedness and inferiority notions 15

course, rint A C % A Consider also the sets

c l A if aff A is a closed set,

1 otherwise,

r i A : = | rint A if aff A is a closed set,

0 otherwise, i6 | l A if XQ is a barreled linear subspace,

- { otherwise,

where X 0 = \in(A — a) for some (every) a G A; XQ is the linear subspace,

parallel to aff A

In the sequel, in this section, A C X is a nonempty convex set Taking

into account the characterization (1.1) of *A, we obtain that

x € tc A <£> cone(A — x) is a closed linear subspace of X

& M X(A — x) is a closed linear subspace of X,

and

x £ A -£> cone(^4 — a;) is a barreled linear subspace of X

<£> M n(A — x) is a barreled linear subspace of X

If X is a Frechet space and aff A is closed then %C A = lb A, but it is possible

to have ib A ^ 0 and ic A = 0 (if aff A is not closed)

The quasi relative interior of A is the set

qri A := {x € A | cone(yl — z) is a linear subspace of X}

Taking into account that in a finite dimensional separated topological vector

space the closure of a convex cone C is a linear subspace if and only if C

is a linear subspace (Exercise!), it follows that in this case qri A = % A =

ic A = ib A

Below we collect several properties of the quasi relative interior

Proposition 1.2.7 Let A C X be a nonempty convex set and T £

L{X,Y) Then:

a £ qri A •& a G A and cone(A — a) = cone(A - A)

O a E i and a - A C cone(yi - a), (1.4)

i A C qriA = AnqriA, V a G q r i A , Vx £ A : [ a , z [ C q r i A , (1.5)

and T(qriA) C qriT(A) In particular qriA is a convex set

Assume that qriA ^ 0; then qriA = A and

*(T(A)) C T(qriA) C qriT(A) C T(A) c T(qriA) (1.6)

Moreover, if Y is separated and finite dimensional then

Proof The first equivalence in Eq (1.4) is immediate from the definition

of qriA Of course, cone (A — a) = cone (A — A) implies that a — A C

cone(A - a) Conversely, if a — A C cone(A - a) then -cone(A - a) =

cone(a — A) C cone (A — a), which shows that cone (A — a) is a linear

subspace Therefore Eq (1.4) holds

If a £ l A, from Eq (1.1) we have that a £ A and cone(A — a) is

a linear subspace, and so cone(A — a) is a linear subspace, too Hence

a € qriA The equality qriA = A n qriA follows immediately from the

relation coneC = cone C, valid for every nonempty subset C of X

Let a € qri A, x e A, A € [0,1[ and a\ :— (1 — X)a + Xx Then

A - A D A - aA = ( l - A)(A - a) + A(A - z) D (1 - A)(A - a),

and so, taking into account Eq (1.4), we have

cone(A - A) D cone(A - a\) D cone ((1 - A)(A - a))

= cone (A — a) = cone (A — A)

Therefore cone(A — A) = cone(A — a^)- Since a\ € A, from Eq (1.4) we

obtain that a\ G qriA The proof of Eq (1.5) is complete

Let a £ qriA; then, by Eq (1.4), a - A C cone(A — a), and so

Ta - T{A) = T(a-A)cT (cone(A - a)) C T (cone(A - a))

= cone (T(A) - Ta), which shows that Ta € qriT(A)

Assume now that qriA ^ 0 and fix a £ qriA It is sufficient to show

Eq (1.6); then the equality qriA = A follows immediately from the last

inclusion in Eq (1.6) for T = Idx and from Eq (1.5)

Let y e i(T'(A)); using Eq (1.1), there exists A > 0 such that (1 + X)y

-XT a € T(A) So, (1 + X)y - -XTa = Tx for some x £ A It follows that

y = Tx x , where x x ~ (1 + A)_1(Aa + x) But, using Eq (1.4), x\ 6 qriA,

and so y 6 T(qriA)

Closedness and inferiority notions 17

The second inclusion in Eq (1.6) was already proved, while the third is

obvious So, let y G T(A); there exists x G A such that y = Tx By Eq

(1.5), (1-X)a + Xx G qriAfor A G]0,1[, and so (l-X)Ta + Xy G T ( q r i A )

for A G ]0,1[ Taking the limit for A —> 1, we obtain that y G T(qri A)

When Y is separated and finite dimensional we have (as already

ob-served) that i (T(A)) = qriT(A); then Eq (1.7) follows immediately from

Eq (1.6) • The notion of quasi relative interior is related to that of united sets Let

X be a locally convex space and A,B C X he nonempty convex sets; we

say that A and B are united if they cannot be properly separated, i.e if

every closed hyperplane which separates A and B contains both of them

The next result is related to the above notions

Proposition 1.2.8 Let X be a locally convex space, A,BcX be

non-empty convex sets and x G X

(i) A and B are united O- cone(A-B) is a linear subspace •& (A — B)~

is a linear subspace

(ii) Assume that cone (A — x) is a linear subspace Then x G c\A

Moreover, i/aff A is closed and rint A ^ 0 then ~x G rint A

Proof, (i) Assume that A and B are united but C := cone(A — B) is

not a linear subspace Then there exists XQ G (—C) \ C By Theorem 1.1.5

there exists x* G X* such that {x 0 , x*) < 0 < (z, x*) for every z G C Then

0 < (x - y,x*) for all x G A, y G B, and so (x,x*) < X < (y,x*) for all

x G A, y G B, for some A G M Therefore H x * t \ separates A and B It

follows that (x,x*) = X = (y,x*) for all x G A, y G B, and so 0 < (z,x*)

for every z G C Thus we have the contradiction (x 0 ,x*) = 0 Therefore

cone (A — B) is a linear subspace

Let C := cone (A — B) be a linear subspace and (x,x*) < X < (y, x") for

all x G A, y G B, for some x* G X* and A e i Then 0 < (z,x*) for every

z G A — B, and so 0 < (z, x*) for z € C Then, since C is a linear subspace,

0 = (z,x*) for every z G C which implies immediately that H x *,\ contains

A and B Therefore A and B are united

The other equivalence is an immediate consequence of the bipolar

the-orem (Thethe-orem 1.1.9)

(ii) By (i) we have that {x} and A are united Assuming that x $ cl A,

using again Theorem 1.1.5, we obtain that {x} and A can be properly

separated This contradiction proves that x E cl A

Suppose now that aSA is closed and rint A / 0 Without loss of

gen-erality we suppose that x = 0 By what was shown above we have that

x = 0 € clA C cl(affA) = aff A Thus X 0 := aS A is a linear space

Assuming now that 0 £ intx0 A ^ 0, we obtain that {x} and A can be

properly separated (in XQ, and therefore in X) using Theorem 1.1.3 This

contradiction proves that x € rint A •

From the preceding proposition we obtain that when X is a locally

convex space and A C X is a nonempty convex set, the quasi relative

interior of A is given by the formula

qri A = A (~1 {x € X \ {x} and A are united}

The next result shows that the class of convex sets with nonempty quasi

relative interior is large enough

Proposition 1.2.9 Let X be a first countable separable locally convex

space and A C X be a nonempty cs-complete set Then qri A ^ 0

Proof Since X is first countable, by Proposition 1.1.11, the topology of

X is determined by a countable family 7 = {p n \ n 6 N} of semi-norms

Without loss of generality we suppose that p n < p n +i for every n € N Since

Ty is semi-metrizable, the set A is separable, too Let A 0 = {x n | n € N} C

A be such that A C cl A 0 Consider j3 n € ]0,2~ n ] such that /3 n p n (x n ) < 2_ n

The series ^n > 1 P n x n is Cauchy (since for m > n and p e N w e have that

PniT^Z^Xk) < ET=mPkPn(x k ) < ET=Z^ Pk (x k ) < 2"™+!) Taking

^n '•= (Z)n>i Pn) 1 Pn, ]Cn>i ^ n1" *s a Cauchy convex series with elements

of A Because A is cs-complete, the series ^n > 1 X n x n is convergent and

its sum x € A Suppose that x ^ qri A Then there exists XQ € (—C)\C,

where C := cb~m(A — x) Using Theorem 1.1.5, there exists x* € X* such

that (xo,x*) < 0 < {z,x*) for all z € C In particular ( *) > 0 for

every x £ A But En> i ^» ( ^ — ^ ^ o ) = 0- Since {x n — X,XQ) > 0 and

A„ > 0 for every n, we obtain that (x - x, x*) = 0 for every x € AQ Since

AQ is dense in A and x* is continuous, we have that {x — x, x*) = 0 for every

x € A, and so (z,a;*) = 0 for every z E C Thus we get the contradiction

Open mapping theorems 19

1.3 Open Mapping Theorems

Throughout this section the spaces X, Y are topological vector spaces if not

stated otherwise We begin with some auxiliary results

Lemma 1.3.1 Let X,Y be first countable topological vector spaces, 31 :

X =$ Y be a multifunction and XQ G X Suppose that grIR satisfies tion (Hwa;) Then

condi-where Nxl^o) ** the class of all neighborhoods of XQ

Proof Let y 0 G C\u£7f x (x 0 )int (clft(t/)) Replacing grft by gv%

-{x 0 ,y 0 ) if necessary, we may assume that (xo,yo) = (0,0) Let U G

Nx-Since X is first countable, there exists a base (U n )n>i C N x of

neighbor-hoods of 0 such that U n + U n C U n -i for every n > 1, where [7o := ?/•

Because 0 G f " ) ^ ^ int (cl#([/)), there exists (Vn)n>i C Ny such that V„ C int (cl3i([7n)) for every n > 1 Since Y is first countable, we may suppose that (V n ) n >i is a base of neighborhoods of 0 G Y and, moreover,

yn +i + V n+ i C KJ for every n > 1

Consider j / ' G int (clCR.(C/i)); there exists fi G]0,1[ such that y := (1 —

l J )~ 1 y' £ cl3?([/i) There exists (£1,2/1) e g r $ such that x\ G C/i and

2/ - 2/i G M^2- It follows that /x-1(y - j/i) G V2 C cl!R(t/2)- There exists

(^2,2/2) G grR such that xi G L/2 and /x_1(2/ _ 2/i) — 2/2 G M^3- It follows that /U~2y — fJ-~ 2 yi — H~ l yi 6 V3 C c\"R{Uz) Continuing in this way we

find ({x m ,y m )) m>1 C grR such that x m e U m and n~ m+x y - n~ m+1 yi

-M-m+22/2 - ••• - y m G fiV m+ i for every m > 1 Therefore um := y 2/1-/^2/2 At",_12/m G ^mK i + i C Vm +i for every m > 1 Moreover,

-Mm_12/m = v m -i -v m e \i m ~ x V m - fi m V m+1 , and so ym £ Vro - fiV m+1 C

V m + V m C Vm_i for m > 2 It follows that ( xm)m> i -> 0, (ym)m>i -> 0 and (um)m>i -> 0, whence J2 m >i n m ~ l y m has sum y Taking Am := ( l - / i ) ^m _ 1, we have that (Am)m>i C R)., Em> i A™ = *> Em> i Am2/m h a s

sum (1 - ju)y = y', the sequence ((xTO,ym)) is bounded (being convergent) and, because £ m t f „+i Ama;TO € C/„+i + U n+2 + ••• + U n+P C f7„+1 + U n+1 C f/n for every n > 1, the series 5 Zm > 1 Amxm is Cauchy By hypothesis there

exists x' £ X, sum of the series ]Cm>i Am#m> such that (x',y') £ g r X Let

x := (1 - n)~ l x'- We have that YZi=i H m ~ lx m G t/i + U 2 + • • • + U n C

t/i -I- Ui C t/ Since U is closed we have that x E.U, and so a;' G (1 — n)U C