# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $$\|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\}$$ How to ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $C(K)$ ...
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### 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? ...
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### 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 ...
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### 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 $$\hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\}$$ ...
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### 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 $e_t \in M$ ...
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### 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, ...
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 ...