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Questions tagged [hahn-banach-theorem]

Questions regarding the Hahn-Banach Theorem.

6
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3answers
793 views

An alternative proof of an application of Hahn-Banach

As a corollary of the Hahn-Banach theorem, we proved that if $M$ is a closed subspace of a normed linear space $X$, $0\neq x_0\notin M$, then $\exists f \in X^*$ such that $f(x_0)\neq 0$ and $f(y)=0$ $...
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1answer
7 views

Question about the range of a norm preserving extension.

Consider the subspace $Y =\{(x, x): x\in \mathbb{C}\}$ of the normed linear space $(\mathbb{C}^2, \|\ \|_{\infty})$. If $\phi$ is a bounded linear functional on $Y$, defined by $\phi(x, x)=x$, then ...
0
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1answer
19 views

Example of weak limit not in the set

Let $(E, \| \cdot \| _E)$ be a normed vector space over a field $\mathbb{K}$ and $E'$ be its dual space. Theorem: Let $C\subset E$ be a closed (respect to the strong topology $\| \cdot \| _E$) and ...
5
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1answer
112 views

Hyperplanes and convex sets in Hilbert space

The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
1
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1answer
25 views

Extension of a linear map in a generic vector space (without Zorn's lemma)

I am studying topological vector spaces from Sevres' book "Topological vector spaces, distributions and Kernels". In one of the preparatory chapters I encountered the following excercise: Consider a ...
8
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1answer
117 views

If the dual of a topological vector space separates points, does it separate a point and a closed subspace?

The Hahn-Banach Theorem implies that if $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, has the following two properties: $X^*$ separates ...
2
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1answer
39 views

Hahn-Banach needed to show equality?

Let $X$ be a normed space and $x_1, x_2 \in X$. Suppose $x^{\ast}(x_1) = x^{\ast}(x_2)$ for all $x^{\ast} \in X^{\ast}$. Then $x_1 = x_2$. Do we need Hahn-Banach (hence, equivalently some sort of ...
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0answers
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Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$

IMPORTANT EDIT: this question has been moved to Mathoverflow It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a ...
0
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1answer
23 views

The extension of functionals

I'm facing the proof, using theorem of Hahn-Banach. The Theorem is following: In normed space every linear, continuous functional f on vector subspace $M \subset X$ can be extended to a continuous ...
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0answers
60 views

Hahn Banach Theorem, First geometric form

I am studying the proof of Hahn Banach Theorem (First geometric form). And the following is a part of the proof. Let $U\subseteq E$ be open, convex and nonempty and let $x_0\in E\backslash U$. Then, ...
2
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1answer
58 views

Corollary of the Hahn Banach theorem

I want to prove the following corollary of the Hahn-Banach theorem. Let $X$ be a normed space. For every closed linear subspace $Y\subseteq X$ and $x\in X-Y$, there exists $x'\in X $ such that $x'|Y=...
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2answers
82 views

Is a Hahn-Banach extension always continuous?

We proved the following version of the Hahn-Banach extension theorem in a course I'm taking: Theorem (Hahn-Banach): Let $X$ be a real vector space and $q : X \to \mathbb{R}$ be sublinear. Let $U \...
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1answer
36 views

are linear functionals on C[0, 1] bounded and thus continuous

I'm currently on a theorem showing a general representation of linear functionals on the space of continuous functions on the interval $[0,1]$. My problem is on the beginning of the proof. First we ...
2
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1answer
77 views

An “open-strict” Version of Hahn Banach Separation Theorem?

Is the following statement true? Let $X$ be a real linear space, $A,B \subset X$ two disjoint convex sets with the following "algebraic openness" property: Every $x \in A$ is an internal point of A, ...
2
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2answers
64 views

(Hahn-Banach) Is a subadditive and positively homogeneous function continuous at $0$?

Let $E$ be topological vector space, i.e., a real vector space over $\mathbb{R}$ such that all points are closed sets and $+,\cdot\,$ are continuous. Let $p: E \to \mathbb{R}$ be a subadditive ($p(x+...
2
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1answer
53 views

How to prove that continuous convex functional on normed vector space must be lower bounded by some continuous affine functional?

Let $X$ be a normed vector space. Let $f:X\mapsto \mathbb{R}$ be a continuous convex function. How to show that there exists a continuous linear functional $l$ and a constant $c\in\mathbb{R}$ such ...
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2answers
38 views

Easy way to extend continous functional

There must be a mistake in my reasoning, but I couldn't find it and that's what I ask you to do: Let $E_0 \subset E$ be a subspace in a normed space $E$ and $f: E_0 \rightarrow \mathbb{R}$ is ...
1
vote
1answer
63 views

Show that $x_0 \in \overline{\langle M \rangle } $ if and only if $f(x_0) = 0, \; \forall f \in X^* : f|_M = 0$.

Exercise : Let $X$ be a normed space and $M \subset X$. Show that an element $x_0 \in X$ belongs to the set $\overline{\langle M \rangle}$ if and only if $f(x_0) = 0$ for all $f \in X^*$ such that $...
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0answers
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Showing that $\exists f \in X^*$ : $\|f\| = \frac{1}{d(x_0,Y)}, \; f(x_0) = 1$ and $f(y) = 0$.

Exercise : Let $X$ be a normed space and $Y$ be a proper closed subspace of $X$. If $x_0 \notin Y$, show that there exists $f \in X^*$ such that : $$\|f\| = \frac{1}{d(x_0,Y)}, \; f(x_0) = 1 \; \...
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2answers
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Proving that $\exists f \in X^*$ : $f(x) = \|x\|^2$ and $\|f\| = \|x\|$

Exercise : Let $X$ be a normed space. Prove that for all $x \in X$ there exists $f \in X^*$, such that $f(x) = \|x\|^2$ and $ \|f\| = \|x \|$. Thoughts : I apologise for not providing a proper ...