Subset of a Banach Space Continuous Functional Separating

Convex Subset

Let K be a convex subset of a topological vector space X and A is a continuous mapping of K into itself so that A(K) is contained in an R-compact subset of K, then A has a fixed point.

From: Random Operator Theory , 2016

TOPOLOGICAL VECTOR SPACES

L.V. KANTOROVICH , G.P. AKILOV , in Functional Analysis (Second Edition), 1982

THEOREM 5.

Let E be a convex subset of an LCS X having non-empty interior

and let F be a non-empty convex subset of X with

∩ F = ∅. Then E and F can be separated. If E and F are open, then they can be strictly separated.

Proof. By the Corollary to Lemma 1,

is convex. Hence the set U =

− F, which is open and does not contain the origin because

∩ F = ∅, is convex. Hence, by Theorem 4, there is a closed real hyperspace H such that 0 ∈ H and (

− F) ∩ H = ∅. Let H = {x: f (x) = 0}, where f ∈ X ^* _R. The set f (

− F) is convex and it is therefore an interval in $\mathcal{R}$ and we have 0 ≠ f (

− F). By a change of sign if necessary, we may assume that f (

− F)<0. Thus sup $\left\{f\left(x\right):x\in \stackrel{\circ }{E}\right\}\le \inf \left\{f\left(x\right):x\in F\right\}.$ . Using the fact that

is dense in E (see the corollary to Lemma 1) and f is continuous, we see that E and F can be separated.

If E and F are open sets, then, by Lemma 3, f (E) and f (F) are open intervals in ℕ, and so the separation is strict.

A closed convex set in an LCS X having non-empty interior is called a convex body. Let E be a subset of an LCS X. A real-valued functional f ∈ X ^* _R is called a supporting functional to E at a point x ₀ ∈ E if there exists λ ∈ R such that f (x ₀) = λ and E is contained in {x: f (x) ≤ λ} or {x: f (x) ≥ λ}. In this situation, the real hyperplane {x: f (x) = λ} is called a supporting hyperplane to E at x ₀.

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Computational Theory of Iterative Methods

In Studies in Computational Mathematics, 2007

Theorem 2.5.2

Let D be a convex subset of a Banach space X and F: D ⊆ X→ Y, If F is Fréchet-differentiable on D and if there exists a constant c such that

(2.5.2) $\left|\left|F^{\prime }\left(x\right)\right|\right|\le M\text{for}allx\in D\Rightarrow \left|\left|F\left(x\right)-F\left(y\right)\right|\right|\le c\left|\left|x-y\right|\right|\text{for}allx\in D.$

The estimate (2.5.1) is the analogue of the famous mean value formula from real analysis. If the operator F′ is Riemann integrable on the segment S we can give the following integral representation of the mean value formula

(2.5.3) $F\left(x\right)-F\left(y\right)={\displaystyle {\int }_0^1{F}^{\prime }\left(x+t\left(y-x\right)\right)dt\left(x-y\right)}.$

Let now D be a convex open subset of X and let us suppose that we have associated to each pair (x, y) of distinct points from D a divided difference [x, y] of F at these points. In applications one often has to require that the operator (x, y) → [x, y] satisfy a Lipschitz condition (see also Section 2.3). We suppose that there exists a nonnegative c > 0 such that

(2.5.4) $\left|\left|\left[x,y\right]-\left[x_1,y_1\right]\right|\right|\le c\left(\left|\left|x-x_1\right|\right|+\left|\left|y-y_1\right|\right|\right)$

for all x, y, x ₁, y ₁ ∈ D with x ≠ y and x ₁= y ₁.

We say in this case that F has a Lipschitz continuous difference on D. This condition allows us to extend by continuity the operator (x, y) → [x, y] to the whole Cartesian product D × D. From (1.2.1) and (2.5.4) it follows that F is Fréchet-differentiable on D and that [x, x] = F′(x). It also follows that

(2.5.5) $\left|\left|F^{\prime }\left(x\right)-F^{\prime }\left(y\right)\right|\right|\le c_1\left|\left|x-y\right|\right|with{c}_1=2c$

and

(2.5.6) $\left|\left|\left[x,y\right]-F^{\prime }\left(z\right)\right|\right|\le c\left(\left|\left|x-z\right|\right|+\left|\left|y-z\right|\right|\right)$

for all x, y ∈ D. conversely if we assume that F is Fréchet-differentiable on D and that its Fréchet derivative satisfies (2.5.5) then it follows that F has a Lipschitz continuous divided difference on D. We can certainly take

(2.5.7) $\left[x,y\right]={\displaystyle {\int }_0^1{F}^{\prime }\left(x+t\left(y-x\right)\right)}dt.$

We now want to give the definition of the second Fréchet derivative of F. We must first introduce the definition of bounded multilinear operators (which will also be used later).

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

10.24 Theorem

Let U be an open convex subset of ℝ ⁿ , and let ϕ: Ux(a, b) → [0, ∞) satisfy (i) ϕ(x, z) is Borel-measureable in z for each fixed x, and (ii) log ϕ(x, z) is convex in x for every fixed z. If v is a measure on the Borel subsets of (a, b) such that ϕ(x, ·) is v-integrable for every x ∈ U, then

(10.46) $\psi (x)\equiv {\displaystyle \underset{a}{\overset{b}{\int }}\phi (x,z)dv(z)}$

is a log-convex function on U.

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Radon-Nikodým Theorems

Domenico Candeloro , Aljoša Volĕiĕ , in Handbook of Measure Theory, 2002

THEOREM 3.13

Let R be any bounded convex subset of [0, ∞[². Then the following properties are equivalent:

(1)

There exists a pair (μ, v) of nonnegative continuous finitely additive measures, defined on a suitable σ-algebra, such that there exists dv/dμ, and such that R is the range of(μ, v).

(2)

R contains its exposed points, and for every segment $L\subset \partial R,$ , at least a (possibly degenerate) closed subsegment I ⊂ L is contained in R. (See Figure 5 (a) and (b).)

Fig. 5. (a) Range closed, (b) Range not closed. Part of the segments is missing.

Usually, if (μ, v) is a pair of nonnegative continuous finitely additive measures, the range looks like in Figure 1, or in Figure 3, when v ≪_ε μ. In Figure 2 a particular situation is shown, where v is obtained by adding to μ some measure which is singular with respect to μ.

Fig. 1. Equation of the lower curve: $y=1-\sqrt{\left(1-x^2\right)}$

Fig. 3. Equation of the Lower curve: y = x ⁶

Fig. 2. Equation of the lower curve: $y=\left(1-\sqrt{\left(1-x^2\right)}\right)/2$ .

We observe that the range is always symmetric with respect to the midpoint (μ(Ω)/2, v(Ω)/2): hence in Figure 5 (a) and (b) just half of $\partial R$ has been drawn.

Fig. 4. Lower curve: $y=\left(1-\sqrt{\left(1-x^2\right)}\right)/2,\text{\hspace{0.17em}for}x<0.5$ , then linear.

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Iterative Methods for Nonexpansive Type Mappings

Abdul Rahim Khan , Hafiz Fukhar-ud-din , in Fixed Point Theory and Graph Theory, 2016

6.1 Introduction and Preliminaries

Many important problems of mathematics including boundary value problems for nonlinear ordinary or partial differential equations can be translated in terms of a fixed point equation Tx  = x for a given mapping T on a Banach space.

The class of nonexpansive mappings contains contractions as a subclass and its study has remained a popular area of research ever since its introduction. The iterative construction of fixed points of these mappings is a fascinating field of research. The fixed point problem for one or a family of nonexpansive (asymptotically nonexpansive) mappings has been studied in Banach spaces and metric spaces [4,20,21,29,35,39,44,47,61–63].

Let C be a nonempty convex subset of a real Banach space E and ℝ be the set of real numbers. A mapping T : C  → C is called:

(a)

nonexpansive if ∥ Tx  − Ty∥   ≤   ∥ x  − y∥ for all x, y ∈ C;

(b)

quasi-nonexpansive if the set F(T)   =   {x ∈ C : T (x)   = x} of fixed points of T is nonempty and ∥ Tx  − y∥   ≤   ∥ x  − y∥ for all x ∈ C and y ∈ F(T);

(c)

asymptotically nonexpansive if there exists a sequence {k_n }   ⊂   [1,∞) with $\underset{n\to \infty }{lim}{k}_n=1$ such that ∥ Tⁿx  − Tⁿy∥   ≤ k_n ∥ x  − y∥ for all x, y ∈ C.

Example 6.1.1

(a)

The usual mappings sin, cos, norm and isometry are nonexpansive.

(b)

T : ℝ   →   ℝ given by $Tx=\frac{x}{2}sin\frac{1}{x}$ and T (0)   =   0 is quasi-nonexpansive with fixed point 0.

(c)

T : ℝ   →   ℝ defined by T(x)   = x ² has two fixed points 0 and 1.

(d)

The translation of a mapping T : ℝ   →   ℝ given by T (x)   = x  + a has no fixed point.

(e)

T : ℝ²  →   ℝ² defined by T(x,y)   =   (x,0) has infinitely many fixed points, i.e., the entire set ℝ.

(f)

The nonself mapping T : [−   1, 1]   →   ℝ given by T(x)   =   1   − x has a fixed point $\frac{1}{2}$ .

(g)

Define T : B  → B by T (x ₁, x ₂, x ₃, …)   =   (0, x ₁ ², a ₂ x ₂, a ₃ x ₃, …) where B is the unit ball in the Hilbert space ℓ ² and {a_i } is a sequence of numbers such that 0   < a_i   <   1 and ${\prod }_{i=2}^{\infty }{a}_i=\frac{1}{2}$ . Then

$||T^{n}x-T^{n}y||\le 2{\displaystyle \prod _{i=2}^n{a}_i||x-y||}\mathrm{f}\mathrm{o}\mathrm{r}n\ge 2\mathrm{and}||Tx-Ty||\le 2||x-y||$

provide that T is asymptotically nonexpansive but not nonexpansive (cf. Ref. [21]).

(h)

Take C  =   [0,1]   ⊂   ℝ and $\frac{1}{2}<k<1$ . For each x ∈ [0,1], we define T : C  → C by

$\begin{array}{lll}\hfill & kx,\hfill & \mathrm{if}0\le x\le \frac{1}{2},\hfill \\ T\left(x\right)=\hfill & -\frac{k}{2k-1}\left(x-k\right),\hfill & \mathrm{if}\frac{1}{2}\le x\le k,\hfill \\ \hfill & 0,\hfill & \mathrm{if}k\le x\le 1\text{.}\hfill \end{array}$

Then T is asymptotically nonexpansive but not nonexpansive (see Ref. [34]).

For sequences, the symbol   →   (respectively ⇀) indicates norm (respectively, weak) convergence. We denote by ω_w (x_n ) the weak ω-limit set of the sequence {x_n }, that is, ${\omega }_w\left(x_n\right)=\left\{x:\exists x_{n_k}\rightharpoonup x\right\}$ . A mapping T : C  → E is demi-closed at y ∈ E if, for each sequence {x_n } in C and each x ∈ E, x_n ⇀ x and Tx_n   → y imply that x ∈ C and T x  = y.

Picard [51] introduced the following iteration formula:

(6.1.1) $x_1\in C,x_{n+1}=T{x}_n\text{.}$

It is well-known that Picard iterations of some nonexpansive mappings fail to converge even on a Banach space. For this, consider an anticlockwise rotation of the unit disc of ℝ² about the origin through an angle of $\frac{\pi }{4}$ . This mapping is nonexpansive but its Picard sequence fails to converge. Krasnoselskii [43] showed that the following iterations formula:

(6.1.2) $x_1\in C,x_{n+1}=\frac{1}{2}\left(x_n+T{x}_n\right)\text{,}$

converges to the fixed point of any nonexpansive mapping T. However, the more general iterative formula for approximation of fixed points of nonexpansive mappings was introduced by Mann [46] as follows:

(6.1.3) $x_1\in C,x_{n+1}=\left(1-{\alpha }_n\right)x_n+{\alpha }_{n}T{x}_n\text{,}$

where 0   ≤ α_n   ≤   1.

Apart from being an obvious generalization of the contraction mappings, nonexpansive mappings are important, as has been observed by Bruck [6], mainly for the following two reasons: (a) Nonexpansive mappings are intimately connected with the monotonicity methods developed in the early 1960s and constitute the first class of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties. (b) Nonexpansive mappings appear in applications as the transition operators for initial valued problems of differential inclusions of the form $0\in \frac{du}{dt}+T\left(t\right)u$ , where the operators {T(t)} are, in general, set-valued and are accretive or dissipative and minimally continuous.

Construction of fixed points of nonexpansive mappings is an important subject in nonlinear mapping theory and has applications in image recovery and signal processing (see, e.g., Refs. [10,52,67]). For the past 30   years or so, the study of Krasnoselskii–Mann iterative procedures for the approximation of fixed points of non-expansive mappings and some of their generalizations and approximation of zeros of accretive-type operators have been a flourishing area of research. For example, the reader can consult the recent monographs of Berinde [1] and Chidume [12]. Mann iterates are not adequate for the approximation of fixed points of pseudocontractive mappings and this led to the introduction of the Ishikawa iterative sequence [25]:

(6.1.4) $x_1\in C,x_{n+1}=\left(1-{\alpha }_n\right)x_n+{\alpha }_{n}T\left(\left(1-{\beta }_n\right)x_n+{\beta }_{n}T{x}_n\right)\text{,}$

where 0   ≤ α_n , β_n   ≤   1.

Note that iterative sequences (6.1.1)–(6.1.3) are special cases of (6.1.4) (through suitable choices of α_n , β_n for all n  ≥   1).

A Banach space E is said to be uniformly convex if, for each ε ∈ (0, 2], the modulus of convexity δ : (0,2]   →   (0,1] of E given by

$\delta \left(\epsilon \right)=inf\left\{1-\frac{1}{2}\left|\right|x+y\left|\right|:\left|\right|x\left|\right|\le 1,\left|\right|y\left|\right|\le 1,\left|\right|x-y\left|\right|\ge \epsilon \right\}$

satisfies the inequality δ(ε)   >   0 for all ε  >   0. It is obvious that $\underset{\epsilon \to 0}{lim}\delta \left(\epsilon \right)=0$ and δ (2)   =   1. We denote the inverse of δ by η and observe that $\underset{y\to 0}{lim}\eta \left(y\right)=0$ . Moreover, it has been shown in [15] that δ is nondeacreasing, continuous and

(6.1.5) $\left|\right|\lambda x+\left(1-\lambda \right)y\left|\right|\le max\left(\left|\left|x\left|\right|,\left|\right|y\right|\right|\right)\left[1-2\lambda \left(1-\lambda \right)\delta \left(\frac{\left|\left|x-y\right|\right|}{max\left(\left|\left|x\left|\right|,\left|\right|y\right|\right|\right)}\right)\right]$

for all x, y ∈ E \ {0} and λ ∈ [0,1].

Let S  =   {x ∈ E : ∥ x∥   =   1} and let E* be the dual of E, that is, the space of all continuous linear functionals f on E.

The space E has:

(a)

Gâteaux differentiable norm [65] if

$\underset{t\to 0}{lim}\frac{\left|\left|x+ty\left|\right|-\left|\right|x\right|\right|}{t}$

exists for each x and y in S;

(b)

Fréchet differentiable norm [65] if for each x in S, the above limit exists and is attained uniformly for y in S and in this case, it has been shown in [65] that

(6.1.6) $〈h,J\left(x\right)〉+\frac{1}{2}\left|\right|x\left|\right|{}^2\le \frac{1}{2}\left|\right|x+h\left|\right|{}^2\le 〈h,J\left(x\right)〉+\frac{1}{2}\left|\right|x\left|\right|{}^2+b\left(\left|\left|h\right|\right|\right)\text{,}$

for all x, h in E, where J is the Fréchet derivative of $\frac{1}{2}\parallel \cdot {\parallel }^2$ at x ∈ E, 〈.,.〉 is the pairing between E and E*, and b is a function defined on [0,∞) such that $\underset{t\to 0^{+}}{lim}\frac{b\left(t\right)}{t}=0$ ;

(c)

Opial's property [50] if for any sequence {x_n } in E, x_n ⇀ x implies that

$\underset{n\to \infty }{limsup}\left|\right|x_n-x\left|\right|<\underset{n\to \infty }{limsup}\left|\right|x_n-y\left|\right|,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{all}y\in E\mathrm{with}y\ne x\text{;}$

(d)

Kadec-Klee property if for every sequence {x_n } in E, x_n ⇀ x and ∥ x_n ∥   →   ∥ x∥ together imply x_n   → x as n  →   ∞.

We state some useful lemmas for later use.

Lemma 6.1.1

cf. Ref. [30]

If {r_n }, {s_n } and {t_n } are non-negative real sequences satisfying

$r_{n+1}\le \left(1+s_n\right)r_n+t_{n}for\mathit{all}n\ge 1,{\displaystyle \sum _{n=1}^{\infty }{s}_n}<\infty \mathit{and}{\displaystyle \sum _{n=1}^{\infty }{t}_n}<\infty \text{,}$

then $\underset{n\to \infty }{lim}{r}_n$ exists.

Lemma 6.1.2

[7]

Let C be a nonempty bounded, closed and convex subset of a uniformly convex Banach space E. Then there is a strictly increasing and continuous convex function g : [0,∞)   →   [0,∞) with g(0)   =   0 such that for a nonexpansive mapping T : C  → E and for all x,y ∈ C and t ∈ [0,1], the following inequality holds:

$\left|\right|T\left(tx+\left(1-t\right)y\right)-\left(tTx+\left(1-t\right)Ty\right)\left|\right|\le g^{-1}\left(\left|\left|x-y\left|\right|-\left|\right|Tx-Ty\right|\right|\right)\text{.}$

Lemma 6.1.3

[26]

Let E be a reflexive Banach space such that E* has the Kadec-Klee property. Let {x_n } be a bounded sequence in E and x*, y* ∈ ω_w (x_n ). Suppose $\underset{n\to \infty }{lim}\parallel t{x}_n+\left(1-t\right)x^{*}-y^{*}\parallel$ exists for all t ∈ [0,1]. Then x*   = y*.

Lemma 6.1.4

[4]

Let C be a nonempty closed and convex subset of a uniformly convex Banach space E and T : C  → C be a nonexpansive mapping. Then I  − T is demi-closed at 0.

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Barrels and Other Features of TVS's

Eric Schechter , in Handbook of Analysis and Its Foundations, 1997

BOUNDED SETS IN ORDERED TVS'S

27.11.

Exercise Let X be an ordered TVS that is locally full (defined in 26.52). Then every order bounded subset of X is toplinearly bounded.

Proof. Let [a, b] be any order interval in X, and let G be any neighborhood of 0 in X. Then G contains some N that is a balanced full neighborhood of 0. Then a, b ∈ rN for r > 0 sufficiently large. Hence [a, b] ⊆ rN ⊆ rG.

27.12.

Theorem Let X be a TVS, and let Y be an ordered TVS that is locally full. Let Ω ⊆ X be open and convex.

If f : Ω → Y is convex and is continuous at some point of Ω, then f is continuous everywhere on Ω.

More generally, let Φ be a collection of convex mappings from Ω into Y. Assume Φ is pointwise toplinearly bounded — i.e., assume that for each x ∈ Ω the set Φ(x) = {f(x) : f ∈ Φ} is toplinearly bounded in Y. Also assume Φ is equicontinuous at some point x ₀ ∈ Ω. Then Φ is equicontinuous at every point of Ω.

Proof (following Neumann [1985]). Let any x ₁ ∈ Ω be given; it suffices to prove equicontinuity at x ₁. We may replace the functions f ∈ Φ with the functions f(⋅ + x ₁) − f(x ₁); thus we may assume that 0 = x ₁ ∈ Ω and that f(0) = 0 for all f ∈ Ω. Let N be any balanced full neighborhood of 0 in Y; we are to show that there is some neighborhood G of 0 in X, contained in Ω, such that ∪ _f∈Ω f(G) ⊆ N.

Choose some balanced full set N′ that is a neighborhood of 0 in Y and satisfies N′ + N′ + N′ ⊆ N. By the assumed equicontinuity at x ₀, there is some balanced neighborhood U of 0 in X, contained in Ω, such that f(x ₀ + u) − f(x ₀) ∈ N′ for all f ∈ Φ and u ∈ U. For some δ ∈ (0, 1] sufficiently small, we have ×δx ₀ ∈ Ω. We know δx ₀ ∈ Ω by convexity of that set. Since Φ is bounded pointwise, B = ∪ _f∈Φ{f(x ₀), f(δx ₀), f(−δx ₀)} is a bounded subset of Y, and hence there is some ε ∈ (0, 1] such that εB ⊆ N′. Since each f ∈ Φ is convex and f(0) = 0, for all u ∈ U we have this estimate:

The vectors on the extreme ends of this estimate belong to N′ + N′ + N′, since N′ is balanced and f(x ₀ ± u) − f(x ₀) ∈ N′. Since N′ + N′ + N′ is contained in N, which is full, we have $f\left(\frac{\epsilon \delta }{1+\delta }\right)\in N$ . Let $G=\frac{\epsilon \delta }{1+\delta }U$ ; this completes the proof.

27.13.

Proposition Let Ω be an open convex subset of a real TVS. Suppose f : Ω, → ℝ is a convex function — or, more generally, suppose that f : Ω → Z is a convex function, where Z is some locally full ordered topological vector space.

Suppose that f is bounded above on some nonempty open set — i.e., there exists some nonempty open set G ⊆ Ω and some z ₀ ∈ Z such that f(x) ≼z ₀ for all x ∈ G.

Then f is continuous. (In particular, any real-valued, upper-semicontinuous, convex function on an open convex set is continuous.)

Proof By translation we may assume 0 ∈ G and f(0) = 0. Replacing G with a smaller open set, we may assume G is balanced. Say f(x) ≼z ₀ for all x ∈ G. Then

so f(x) ≽ −z ₀ for all x ∈ G. Thus f is order bounded on G.

Let any positive integer n be given. If $x\in \frac{1}{n}G$ , then nx ∈ G, so

Thus f is bounded above by $\frac{1}{n}{z}_0$ on $\frac{1}{n}G$ . By the argument of the preceding paragraph, f is bounded below by $\begin{array}{lll}-\frac{1}{n}{z}_0\hfill & \text{on}\hfill & \frac{1}{n}G\hfill \end{array}$ . By the Squeeze Property (26.52(E)), it follows that lim _x→0 f(x) = 0; thus f is continuous at 0. By 27.12, f is continuous everywhere on Ω.

Corollary Suppose Ω is an open convex subset of ℝ ⁿ . Then any convex function f : Ω → ℝ is continuous.

Proof of corollary For any x ∈ Ω, let N(x) be a closed n-dimensional cube centered at x, small enough to be contained in Ω. That cube has 2 ⁿ vertices v ₁, v ₂, …, v _{2 ⁿ} . Each point u in N(x) is a convex combination of the v _j 's, and so sup _u∈N(x) f(u) ≽ max _j f(v _j ).

Remark A convex function on an infinite-dimensional normed space is not necessarily continuous; see 23.6.a and 23.6.b.

27.14.

Proposition Let X be a topological vector space whose topology is given by an F-norm. Let Ω be an open convex subset of X. If f : Ω → ℝ is convex and continuous, then f is locally Lipschitz.

Proof Suppose that the topology on X is given by an F-norm ρ. Let any point x ₀ ∈ Ω be given; we shall show f is Lipschitzian on some neighborhood of x ₀. (This argument is based on Roberts and Varberg [1974].) Let B _r denote the open ball of radius r centered at x ₀. Choose r > 0 small enough and M large enough so that B _2r ⊆ Ω and f(⋅) ≤ M on B _2r and -f(x ₀) ≤ M; we shall show f is Lipschitzian on B _r with 〈f〉_Lip ≤ 4M/r. Note that for any u ∈ X with ρ(u) < 2r, we have

$\begin{array}{lllllllll}-M\hfill & \le \hfill & f\left(x_0\right)\hfill & =\hfill & f\left(\frac{\left(x_0+u\right)+\left(x_0-u\right)}{2}\right)\hfill & \le \hfill & \frac{f\left(x_0+u\right)}{2}+\frac{f\left(x_0+u\right)}{2}\hfill & \le \hfill & \frac{f\left(x_0+u\right)}{2}+\frac{M}{2}\hfill \end{array}$

and therefore f(⋅) ≥ −3M on B _2r .

Let any distinct points x ₁, x ₂ ∈ B _r be given; let α = ρ(x ₁ − x ₂). Let $x_3=x_2+\frac{r}{\alpha }\left(x_2-x_1\right)$ note that x ₃ ∈ B _2r . Compute

$\begin{array}{lllll}f\left(x_2\right)\hfill & =\hfill & f\left(\frac{r}{r+\alpha }{x}_1+\frac{\alpha }{r+\alpha }{x}_3\right)\hfill & \le \hfill & \frac{r}{r+\alpha }f\left(x_1\right)+\frac{r}{r+\alpha }f\left(x_3\right)\hfill \end{array}$

and thus

$f\left(x_2\right)-f\left(x_1\right)\le \frac{\alpha }{r+\alpha }\left[f\left(x_3\right)-f\left(x_1\right)\right]\le \frac{4M\alpha }{r+\alpha }\le \frac{4M\alpha }{r}=\frac{4M}{r}\rho \left(x_1-x_2\right).$

Similarly, $f\left(x_1\right)-f\left(x_2\right)\le \frac{4M}{r}\rho \left(x_1-x_2\right).$ .

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Convex Programming

Vadim Azhmyakov , in A Relaxation-Based Approach to Optimal Control of Hybrid and Switched Systems, 2019

3.2 Existence Theorems

The general existence result was mentioned in Section 2.2 (Theorem 2.17). The corresponding geometrical characterization of the set $\mathcal{F}$ of all optimal solutions to (2.1) is also well known (see, e.g., [198,275]).

Theorem 3.6

Let Q be a nonempty convex subset of a real Hilbert space H. For every quasiconvex functional $J:Q\to \mathbb{R}$ the set of minimal points of $J(\cdot )$ on Q is convex.

Evidently, Theorem 3.6 can also be applied to problem (2.1). Theorem 3.6 can be easily proved by a direct calculation. It is necessary to underline that the result of Theorem 2.17 is an immediate consequence of the following general existence result for real Banach spaces.

Theorem 3.7

Let Q be a nonempty, convex, closed, and bounded subset of a reflexive real Banach space, and let $J:Q\to \mathbb{R}$ be a continuous quasiconvex functional. Then problem $J(\cdot )$ has at least one minimal point on Q.

Recall (see Chapter 2) that if we eliminate the boundedness hypothesis from Theorem 3.7 we need to make some additional assumptions in order to guarantee the existence of an optimal solution in problem (2.1). Evidently, Theorem 2.30 makes it possible to establish the existence of an optimal solution to (2.1) in simple terms of a minimizing sequence.

The next fundamental result constitutes in some sense a helpful tool for the existence proofs (see, e.g., [6]). Let us note that this result is generally true in a reflexive Banach space.

Theorem 3.8 Eberlein–Smulyan theorem

Let H be a real Hilbert space and

$B_0:=\{x\in H:||x|{|}_H\le 1\}$

be the closed unit ball. Then the following are equivalent:

•

$B_0$ is weakly compact;

•

$B_0$ is sequentially weakly compact;

•

every bounded sequence in H has a weakly convergent subsequence.

As mentioned in Section 3.1 a practically oriented (engineering) "solution" to the general convex minimization problem (2.1) is represented by a minimizing sequence. In a "worst case" the minimizing sequence generated by a numerical optimization method does not possess any convergence property. Since the concept of a minimizing sequence can be considered as a new numerically oriented solution concept, we are interested in the existence of a convergent minimizing sequence for the original problem (2.1). So we also understand in this book an "existence question" from the point of view of numerical analysis and study certain criteria under which a minimizing sequence must converge in norm. Such conditions must be known before the iterative and extrapolative numerical solution procedure and can be used in the computational step for the approximate minima search. Note that the class of applied problems which can be effectively interpreted as minimization problems is growing rapidly and the numerical solution concept, namely, a generation of a numerically stable minimizing sequence, constitutes nowadays a challenging engineering and mathematical problem.

Taking into consideration the above motivating arguments we next present some general results related to the convergent minimizing sequences. First of all let us study a counterexample.

Example 3.1

Consider the specific case $H=l^2$ and a closed positive cone C in H. Then the set $Q:=C\bigcap B_0$ , where $B_0$ is a closed unit ball centered at zero, is closed and convex. By Theorem 3.8 it is also weakly compact. Let

$e_1=(1,0,O,...),e_2=(0,1,O,...),...$

be elements of the canonical orthonormal Schauder basis. That means every $x\in l^2$ can be expressed as

(3.1) $x=\sum _{i=1}^{\infty }{\alpha }_i{e}_i.$

Consider

$y=(1,1/2,1/3,...)\in l^2$

and define a linear functional

$J(x):=\sum _{i=1}^{\infty }{\alpha }_i(1/i)$

for $x\in l^2$ defined by (3.1) . Since $J(\cdot )$ has a unique minimum at $x=0$ , the sequence

$\{e_i\},i\in \mathbb{N},$

is a minimizing sequence for problem (2.1) . We have

$J(e_i)=1/i\to 0=J(0).$

However,

$||e_i-e_j|{|}_{l^2}=2^{1/2},i\ne j,$

so that $\{e_i\},i\in \mathbb{N}$ , contains no subsequences which converge to 0 in norm.

Note that Example 3.1 shows that a minimizing sequence to the problem (2.1) does not necessarily contain any norm-convergent subsequence.

We next will see that the usual first-order numerical optimization methods generate a weakly convergent minimizing sequence (Section 3.6). The next general result illustrates a possibility to obtain a strongly convergent sequence from a weakly convergent one. This analytic result is also true in general normed spaces (see, e.g., [192]).

Theorem 3.9 Mazur lemma

Assume H is a real Hilbert space, and assume $\{x_k\},k\in \mathbb{N}$ , is a sequence converging weakly to $x\in H$ . Then there is a sequence

$\{x_k\},k\in \mathbb{N},$

of convex combinations of $\{x_k\},k\in \mathbb{N}$ , which converges strongly to $x\in H$ .

Let us also recall that the extremal set of a compact convex subset of a reflexive Banach space X need not be compact.

Theorem 3.10 Krein–Milman

Let X be a locally convex Hausdorff topological vector space and let $Q\subset X$ be a nonempty compact convex set. Then Q is the closed convex hull of its extremal points $Ext(Q)$ , i.e.,

$Q=\overline{conv}(Ext(Q)).$

In particular, Q admits an extremal point, i.e., $Ext(Q)\ne \varnothing$ .

Assume now X is strictly convex, i.e., for all $x,y\in X$ ,

$||x+y|{|}_X=2||x|{|}_X=2||y|{|}_X\Rightarrow x=y.$

Recall that a real Hilbert space H is a reflexive and strictly convex Banach space [20]. The closed unit ball $B_0$ of H is weakly compact by Theorem 3.8 and its extremal set is the unit sphere $S(B_0)$ . Thus the extremal set is not weakly compact and $B_0$ is the weak closure of its extremal set.

We finally discuss the convergence results for minimizing sequences in convex optimization. If $J(\cdot )$ has certain analytical properties then convergence in H-norm is guaranteed for any minimizing sequence. It is well known that in the case $J(\cdot )$ is strictly quasiconvex and continuous on a compact convex subset Q of a real Hilbert space H, $J(\cdot )$ is uniformly quasiconvex on Q. A direct consequence for problem (2.1) of this geometrical fact can be expressed as follows (see [131]).

Theorem 3.11

If $J(\cdot )$ is continuous and strictly quasiconvex on a compact convex set Q, then any minimizing sequence for (2.1) converges in H-norm to a unique minimum.

We next need a relative fine analytic concept, namely, the concept of a dentable set [276]. Let

$\mathrm{epi}\{J(\cdot )\}$

be the epigraph of $J(\cdot )$ on Q.

Definition 3.4

We will say that $J(\cdot )$ is dentable on Q at $x_0$ , whenever, for $ϵ>0$ , the point $(x_0,J(x_0))$ does not belong to the closed convex hull of

$\mathrm{epi}\{J(\cdot )\}\setminus B_{ϵ},$

where $B_{ϵ}$ is the ϵ-ball centered on $(x_o,J(x_0))$ .

Theorem 3.12

Let $J(\cdot )$ be a lower semicontinuous convex functional on a weakly compact convex subset of a Hilbert space H with a unique minimum $x_0\in Q$ . If $J(\cdot )$ is dentable at $x_0$ , then any minimizing sequence

$\{x_k\}\subset Q,k\in \mathbb{N},$

converges in H-norm to $x_0$ .

Note that the proof of this result is based on the generic properties of the projection operator and also uses the fundamental separation principle.

Theorem 3.13

Assume $J(\cdot )$ is strictly quasiconvex and lower semicontinuous on a weakly compact convex set $Q\subset H$ and is dentable at its unique minimum. Then any minimizing sequence

$\{x_k\},k\in \mathbb{N},$

for (2.1) converges in H-norm to the minimum.

One can show that in Example 3.1

$\inf \{J(x):x\in Q\}=0$

but the point $(0,0)$ is not a denting point of $\mathrm{epi}\{J(\cdot )\}$ . The necessary conditions for a convergence of a minimizing sequence are also closely related to the concept of dentable sets (Definition 3.4).

Theorem 3.14

Assume $J(\cdot )$ is a lower semicontinuous quasiconvex functional on a closed convex $Q\subset H$ and suppose that (2.1) has a unique solution

$x_0\in Q.$

If every minimizing sequence of (2.1) converges to $x_0\in Q$ in H-norm, then $J(\cdot )$ is dentable at $x_0$ .

We finally give the following general result.

Theorem 3.15

Let $J(\cdot )$ be a lower semicontinuous strictly quasiconvex functional on a weakly compact convex subset Q of a Hilbert space H. Then every minimizing sequence for (2.1) converges in H-norm to a unique minimum $x_0\in Q$ iff $J(\cdot )$ is dentable at $x_0$ .

Evidently, a convex and closed set Q in a real Hilbert space H is weakly compact (see Theorem 3.8 above). Without the weak compactness there may be no minimum in (2.1). Of course, even in that case a minimizing sequence

$\{x_k\},k\in \mathbb{N},$

for (2.1) can exist.

Example 3.2 K. Weierstrass

Let

$Q:=\{g(\cdot )\in {\mathbb{L}}^2([t_0,t_f],R):g(-1)=-1,g(1)=1\}\bigcap {\mathbb{C}}^1(-1,1),$

where ${\mathbb{L}}^2([t_0,t_f],R)$ is the Lebesgue space of all square integrable functions from $[t_0,t_f]$ to $\mathbb{R}$ .

Let $J:Q\to \mathbb{R}$ with

(3.2) $J(g(\cdot )):={\int }_{-1}^1{t}^2{[g(t)]}^{2}dt,$

where $g(\cdot )$ is a derivative of the function $g(\cdot )\in$ . The infimum of $J(\cdot )$ on the convex set Q is equal to 0. Since the integrand in (3.2) is nonnegative, we get

$J(g(\cdot ))>0$

for every $g(\cdot )\in Q$ . Thus there is no minimum of $J(\cdot )$ on Q. However,

$g_k(t):=\arctan (kt)/\arctan (k)$

defines a minimizing sequence for the resulting problem (2.1) ;

$J(g_k(\cdot ))<{\int }_{-1}^1(t^2+(1/k)){(g(t))}^{2}dt=2/k\arctan (k)\to 0$

for $k\to \infty$ .

In the above example the convex set Q is nonclosed (in the sense of the ${\mathbb{L}}^2([t_0,t_f],R)$ -norm) and is nonweakly compact.

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The Classical Stefan Problem

In North-Holland Series in Applied Mathematics and Mechanics, 2003

Proposition 10.1.5

W is continuous on the closure $\overline{V}$ of V. W maps $\overline{V}$ into itself.

Since $\overline{V}$ is a compact and convex subset of a Banach space, and W is a continuous mapping of $\overline{V}$ into itself, by Schauder's fixed point theorem [282], there exists at least one element of $\overline{V}$ which is left invariant under W, i.e., S ⁰(t) = W(S ⁰(t)). The function S ⁰(t) is the solution of (10.1.11) and T(x, t;S ⁰) is the solution of reduced problem (10.1.8)–(10.1.10).

It can be proved that if S(t) is in $\overline{V}$ and S(t) = W(S(t)), then S(t) is differentiable, i.e., S(t) is in V. Further

$\frac{dS}{dt}=b-{\left(p{T}_x\right)|}_{x=S\left(t\right)},$

and therefore (S(t), T(x, t; S(t))) is a solution of the problem (10.1.1)–(10.1.4). Uniqueness of the problem (10.1.1)–(10.1.4) has also been proved in [279] but it is not based on contraction mapping argument [282].

The problem considered in [280] can be obtained by making some changes in (10.1.1)–(10.1.4), such as, take p = 1, f = 1, b = 0, S(0) = A and T(x, 0) = a(x), where 0 < a(x) < d(A - x), 0 < x < A, and d is some positive constant. The functions a(x) and g(t) are continuous, g(t) < d. An integral equation of the type (10.1.7) has been obtained in this case also and now a reduced problem of the type (10.1.8)–(10.1.10) will have a prescribed initial temperature also as S(0) = A > 0. The existence of the solution on some finite time interval [0,t] has been proved using fixed point theorem as in [279]. However, the uniqueness of the solution has been proved by showing that the iterations done in the numerical solution for calculating the free boundary are converging, i.e., if S ₀ = A and S _n+1 = F(S_n ), then F is a contraction.

For the numerical solution of the problem considered in [280], the time interval [0, t] is divided into n small time intervals, each of length δt and in each one of them iterations are done to get better values of S(t). For this purpose an integral equation of the form (10.1.11) is used. T(x, t;S(t)) is obtained from the solution of the 'reduced problem' formulated for this problem with appropriate changes. The first iterative process will converge to the solution if the time interval is small (existence holds and uniqueness is proved by contraction argument). Then another iterative process is carried out in the time interval (0,2δt]. The solution of the previous time step is used to obtain S(t) and T(x, t;S(t)) in (0,2δt] by using a suitable iterative process (cf. [280]). The initial temperatures in the 'reduced problems' will go on changing. This procedure is repeated in other time intervals also till the solution is obtained in the time interval [0, t]. It has been shown that $\underset{n\to \infty }{lim}{S}_{n+1}=S\left(t\right)$ and S(t) is invariant under the transformation of the form (10.1.12) for the present problem also. The subscript n refers here to the n-th iterative process.

The problem considered in [281] can be obtained from (10.1.1)–(10.1.4) if we take p = 1 and f = 1 in (10.1.1), b = 0 in (10.1.4), S(0) = d and replace (10.1.2) by a temperature prescribed boundary condition, e.g.,

(10.1.15) $T\left(0,t\right)=v\left(t\right)\ge 0,0<t<t_{*}<\infty .$

Since S(0) = d, initial temperature is to be prescribed and let

(10.1.16) $\begin{array}{cc}T\left(x,0\right)=T_0\left(x\right)\ge 0,& 0\le x\le d.\end{array}$

We shall refer to this problem in [281] as Problem (F). The main result of [281] is the following proposition.

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The Riesz Theorem

Joe Diestel , Johan swart , in Handbook of Measure Theory, 2002

THEOREM 5.1 (Bishop-de Leeuw)

If K is a non-empty compact convex subset of a locally convex Hausdorff linear topological space E, then each point x of K is the barycenter of a regular Bore1 probability measure μ on K that's supported by the extreme points of K's in the sense that μ (B) = 0 for anj Baire subset B of X that contains none of K 's extreme points.

G. Edgar [43] surprised the mathematical world when he provided a generalization that relaxed the compactness condition.

THEOREM 5.2 (Edgar)

Let K be a non-empty closed bounded convex subset of the Banach space X. Suppose K is separable and has the Radon-Nikodým propert): Then each point is the barycenter of a regular Borel probability measure supported by the set of K 's extreme points.

Separability plays the role of providing access to the methods and results of descriptive set theory, most particularly, the Kuratowski/Ryll-Nardzewski selection theorem; the Radon-Nikodým property enters the foray through ingenious use of the Martingale Convergence Theorem. The monograph of R. Bourgin [I51 should be consulted here.

Another fascinating effort at generalization (albeit in a different direction) was expanded by S. Khurana [61, 62] whose aim was a finitely additive representation theory. As yet, this promising approach has been just that: promising.

The Choquet theorem has been the inspiration of several brilliant treatises. First, R.R. Phelps' still-wonderful pocket-book [73] is a must; it's too bad it's out-of-print. Next, E.M. Alfsen's monograph [1] presents the subject in a clear and compelling manner. Finally, the master himself puts his theorem (and, oh, so much more) into perspective in his treasured three volumes [20–22].

Rainwater's theorem is due, of course, to the ubiquitous John Rainwater [80].

It has descendants of outstanding pedigree, to be sure. John Elton [44] showed the following.

THEOREM 5.3 (Elton)

In order that every series ${\displaystyle {\sum }_n{x}_n}$ in a Banach space X for which ${\displaystyle {\sum }_n\left|x*(x_n)\right|<\infty }$ for each x* ∈ ex(B_X*) be unconditionally convergent it is both necessary and suficient that X contains no isomorphic copy of c₀ .

Be forewarned: if you recall the classic result of C. Bessaga and A. Pełczyński [9] which asserts that it's precisely in spaces X without copies of c ₀ ${\displaystyle {\sum }_n{x}_n}$ that is unconditionally convergent when ${\displaystyle {\sum }_n\left|x*(x_n)\right|<\infty }$ for each x* ∈ X*, then be aware that Elton's Theorem 5.3 is considerably more subtle, relying on ideas related to the Radon-Nikodým property. In particular, adroit use is made of another classical result of Bessaga and Pdczyński [10] this time to the effect that in separable duals every non void closed bounded convex set has extreme points and is the closed convex hull of the set of such.

J. Bourgain and M. Talagrand [14] also entertained questions involving extremal convergence.

The applications of the Riesz Theorem in tandem with the Hahn-Banach theorem often involve extreme point considerations. Nowhere is this more powerfully demonstrated than in de Branges's proof of the Stone-Weierstrass theorem. This paper [25] was somewhat typical of many beautiful proofs from the 1960's involving function algebras. The monographs of T. Gamelin [47] and L. Stout [96] provide clear exposition of many of the best results of the times.

De Branges's proof still has a lot of kick in it. Most recently, Victor Lomonosov [66] used ideas implicit in de Branges's proof to show that there exists a uniform algebra A and a non-void closed bounded convex subset K of A such that the set of members of A* that attain their maximum modulus on K is not norm dense in A*. Thus one of the most beloved consequences of the Bishop-Phelps theorem is not so in complex Banach spaces.

Now Lomonosov is just the kind of mathematician who knows a good thing when he sees it and soon he turned his counter-example into a positive result [67] showing that all dual uniform algebras in which the fill Bishop-Phelps result holds are self-adjoint. This is startling mix of algebraic and geometric thinking, and the Riesz Theorem is again at the center of the action.

A. Pietsch's stunning Domination Theorem appeared in the paper [74] in which he first introduced the class π_p of absolutely p-summing operators. Our proof is a small modification and special case of an argument of Bernard Maurey [68].

Pietsch's isolation of the absolutely p-summing operators was a singular event in abstract analysis in general and Banach space theory in particular. It's true that in one guise or another Grothendieck recognized π₁, and π₂, but in the latter case he did not have the same norm as Pietsch did. This is important, as witnessed by the remarkable result discovered by D.J.H. Garling and Y. Gordon [48], and, independently. M.í. Kadec' and M.G. Snobar [58]: if E is a finite dimensional Banach space, then ${\pi }_2(i{d}_E)=\sqrt{\dim E}$ . Precise computations like this are frequently encountered in the literature on absolutely summing operators, particularly as they relate to other aspects of mathematical enquiry. We cannot overstate our respect for Pietsch's accomplishment. Our attitude is buttressed by the many monographs that center on the Pietsch theorem and its ramifications. To be precise let us mention: [72] (where connections are made with spaces of analytic functions), [77] (which shows the role played by factorization schemes in harmonic analysis and operator theory), [63, 75] (each of which draws deep conclusions about eigenvalue distributions for classical operators from the theory of p-summing operators and their relatives), [99] (where Minkowski spaces are the object of attention from a π _p -perspective). [78] (which studies volume ratios and the geometry of convex bodies with a π _p -helper) and [76] (where classical studies on orthonormal systems are given a refreshing new look). In addition the Cambridge books of [30] and [102] give ample overview to how summing operators 'fit' within the fabric of abstract analysis.

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The Bishop–Phelps Theorem

R.R. Phelps , in 10 Mathematical Essays on Approximation in Analysis and Topology, 2005

8 Miscellany

As noted in the introductory section, a closed convex subset C of a Banach space can always be represented as the intersection of the closed half–spaces which support it. The question as to which sets S of support points can be removed from C and still have C represented by the half spaces supporting it at C\S has been investigated in [28]. The question as to whether vector–valued lower semicontinuous convex functions need have subdifferentials was answered in the negative in [29]; this is a consequence of an example of two proper lower semicontinuous functions on ℓ ₂ with no common point of subdifferentiability.

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