0
votes
1answer
37 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
3
votes
1answer
28 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
1
vote
0answers
50 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
0
votes
0answers
35 views

proof of the existence of spherical sections of ellipsoids

i want to prove : Let L be proper ellipsoid with the origin as center in $E^{2m-1}$ .There exists a subspace $E^m$ such that $E^m$ intersects $L$ is an m-dimensional sphere it is proven by Dvoretzky ...
0
votes
1answer
26 views

Why is it a borel set on the boundary of the unit ball of $E^n$?

Given $C$ convex body (compact convex set with non-empty interior points) in $E^n$ symmetric about the origin and containing the unit ball. Let $A(r)$ denote ,for every real $r >1$, the subset of ...
1
vote
1answer
36 views

what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set ...
0
votes
1answer
32 views

Show that $L^p$ norm is logarithmically convex as a function of $p$

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded domain. Let $u : \Omega \to \mathbb{R}$ be measurable with $||u||_\infty < \infty$. For $p \in [1, \infty)$, define $$\Phi_u : p \mapsto ...
2
votes
1answer
55 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
0
votes
1answer
23 views

Continuity of bounded and convex function on Hilbert space

I'm looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some ...
0
votes
1answer
19 views

separate convex and concave function by affine function

Let $f:\mathbb R^n\to\mathbb R$ be a concave continuous function and $g:\mathbb R^n\to\mathbb R$ be a convex continuous function such that $f\leq g$. Then there exists an affine continuous function ...
0
votes
0answers
38 views

Hahn-Banach separation theorem

Let $V_1,V_2$ be convex subsets of a normed space $X$ with $V_1^\circ\neq\emptyset$ and $V_1^\circ\cap V_2 =\emptyset$. Then there exists $x'\in X'\setminus\{0\}$ such that $$\operatorname{Re} ...
1
vote
1answer
19 views

Infinite dimensional convex cones

Let $C$ be a convex cone in a topological real vector space $V$. Assume that we have a linear functional $\varphi: V \to \mathbb{R}$ such that $\varphi(x) \geq 0$ for all $x \in C$. Further assume ...
0
votes
1answer
44 views

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X'$ be strict convex, i.e. for all $x_1',x_2'\in X'$ with $\|x_1'\|_{X'}=\|x_2'\|_{X'}=1$ the implication $$\left\|\frac{x_1'+x_2'}{2}\right\|=1\Rightarrow x_1'=x_2'$$ holds. In this case the ...
2
votes
0answers
35 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
0
votes
0answers
21 views

Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
2
votes
1answer
44 views

A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
2
votes
0answers
118 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
0
votes
1answer
22 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
0
votes
1answer
24 views

Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...
0
votes
0answers
26 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
0
votes
1answer
28 views

Are there necessary and sufficient conditions for Krein-Milman type conclusions?

This, the third of three self-answered questions, contains a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The second question is here. ...
0
votes
1answer
14 views

Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?

This is the second of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The third ...
1
vote
0answers
66 views

When do partial subgradients give a subgradient?

I'm looking for sufficient conditions that guarantee that partial subgradients of a convex, lower-semicontinuous functional $f:X_1\times X_2\rightarrow\overline{\mathbb{R}}$ form a subgradient of $f$. ...
0
votes
1answer
30 views

Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?

This is the first of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The second question is here. The third ...
1
vote
1answer
66 views

Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
0
votes
1answer
16 views

Establishing convexity of a function

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and ...
2
votes
0answers
128 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
4
votes
2answers
81 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
4
votes
0answers
222 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
1
vote
1answer
55 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
1
vote
0answers
31 views

The equivalent definition of denting point

How i can prove that If $K$ is a subspace of Banach space $X$, $x$ is denting point of $K$,when for every $\varepsilon>0$,there is a unit vector $x^{*}\in X^{*}$ and $\delta>0$ such that ...
1
vote
1answer
62 views

When is $f(X)$ convex?

Let $X$ be a Banach space, and let $f: X \to X$ be a nonlinear operator, $\mathrm{Dom}(f)=X$. When is $f(X)$ convex?
2
votes
0answers
47 views

Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
1
vote
0answers
26 views

can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
0
votes
1answer
52 views

There is a closed hyperplane.

$\textbf{Question: }$ If $M$ is an open convex set in normed linear space $R$ and $x_{0}\not\in M$, then there exists a closed hyperplane which passes through the point $x_{0}$ and does not intersect ...
0
votes
1answer
35 views

Space where the separation theorem doesn't hold

I have read the proof of the next separation theorem: Let X b a normed space in R and A a convex and open set that contains 0. Let b be a point wich is not in A then there exists f in X* such that ...
3
votes
1answer
31 views

nonsurjectivity of Banach-Stone theorem

Currently I'm studying the extension of Banach-Stone theorem using this book. There is one section about the removal of surjectivity of the isometric isomorphism between the two function spaces $C(K)$ ...
2
votes
2answers
87 views

Convex Sets in Functional Analysis?

Why is it that convex sets and convex functions are a) so important & b) so intrinsically related to functional analysis as to deserve an entire chapter in Bourbaki's topological vector spaces? ...
3
votes
2answers
67 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
2
votes
1answer
45 views

What is the name of this object?

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let $$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$ ...
1
vote
1answer
56 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
1
vote
1answer
60 views

Property for the subdifferential and duality mapping in context of the Moreau-Yosida regularization

I have a question arising from the Moreau-Yosida regularization in Banach spaces. The real Banach space $X$ and its dual $X^*$ are both reflexive strictly convex, $f:X \rightarrow \mathbb{R} \cup ...
0
votes
1answer
56 views

Prove range of f',$\{f'(x),x\in X\}$ dense in $X^*$

Let $X$ be a Banach Space and let $f: X\rightarrow \Bbb R$ be a Fre'chet differentiable function. Suppose that $f$ is bounded from below on any bounded set and satisfies $lim_{||x||\rightarrow ...
1
vote
0answers
55 views

Convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$

I want to prove the convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$ and here is what I've done so far: Since $f$ is convex, $f(\frac{dP}{dQ})$ is also convex w.r.t. $dP$ because $dQ$ ...
5
votes
1answer
130 views

Is this set convex?

I have been trying to show that the following set is convex, with no luck. I am not even entirely convinced that it is in fact convex. A small hint would be greatly appreciated. For $M>0:$ $$ ...
1
vote
1answer
44 views

average of a bounded convex set

Suppose $X$ is a bounded convex set. We know that the average of any $n$ points of $X$, belongs to it, i.e. if $x_1, x_2, . . . , x_n \in X$ then $\frac{x_1+x_2+\cdots +x_n}{n}\in X$. How can we ...
3
votes
1answer
70 views

What type of convex constraint is defined by SQRT?

Let $A$ be an $n \times n$ positive semidefinite matrix and $\forall k, x_k \in \mathbb{R}^n$. The distance with respect to this matrix is defined as $ \|x_i -x_j\|_A := ...
0
votes
0answers
25 views

Finding Borel measures on a closed convex hull

Let $M=C(I)^{\ast}$, the space of complex Borel measures on the unit interval $I$. Suppose we give $M$ the weak*-topology induced by the Banach space $C(I)$. Now $\forall$ $t \in I$, let $e_t \in M$ ...
0
votes
1answer
63 views

How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
4
votes
2answers
173 views

Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...