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Questions tagged [hahn-banach-theorem]

Questions regarding the Hahn-Banach Theorem.

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Applying Geometric Hahn-Banach Theorem on Vector-Valued Integration

I have been studying the applications of geometric Hahn-Banach Theorems. I came across this book Functional Analysis by S. Kesavan where the geometric Hahn-Banach Theorem is applied on Vector-Valued ...
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30 views

Generalized limit in $\mathcal{L} _{\infty}$ (Using: Hahn Banach Extension Theorem)

I am trying to proof the same as the question bellow, but for bounded functions over a field $\mathbb{K}$. p is defined the same way as the question but we are taking the limit of a function when it's ...
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1answer
55 views

Why is this a corollary from Hahn Banch?

On Wikipedia I found that the following is a corollary of Hahn Banach: Let $X$ be a normed vector space. $M \subset X$ a linear subspace and $x_0 \in X$ such that $d:=\inf\limits_{y\in M}||x_0-y|| &...
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50 views

Hahn-Banach Theorems Applications

Please, if anyone can help me with some useful tips to solve this aim: Let $K^1,...,K^n$ closed convex sets containing the origin of a normed space $E$, and let $c_1,...c_n$ positive real numbers. ...
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41 views

$M \leq X$. Is it true that $M^*$ can be isometrically embedded in $X^*$ where $*$ denotes the dual space?

Let $X$ be a Banach space and $M$ be a closed subspace i.e. $M \leq X$. Is it true that $M^*$ can be isometrically embedded as a subspace of $X^*$ where $*$ denotes the dual space? What I have tried:...
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33 views

prove a set is non empty using Hahn-Banach theorem

Let $(E, ||.||_E )$ be a banach space, and its $E^*$ topological dual. For $u ∈ E, F(u) =\{L\in E^*, ||L||_{E^∗} = ||u||_E, \left<L, u\right> = ||u||^2_E \}$ Prove $F(u)$ is non empty and ...
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16 views

prove unicity of a linear continous form

Let H be a Hilbert space and $ u ∈ H$. Prove there exists a unique continuous linear form $L∈ H^*$ such that: $||L||_{H^∗} = ||u||_H$ and $<L, u> = ||u||^2_H$ I proved the existence : We can ...
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45 views

Existence of a linear functional. Hahn-Banach.

Given a sublinear functional $p$ in a real vector space $X$, show that there exists a linear functional $f$ in $X$ such that $-p(-x)\leq f(x)\leq p(x)$. I am trying to use the Hahn-Banach theorem ...
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68 views

Approximating a Banach space “vector- valued function” by “simple functions”

I'm trying to prove the following claim: Let $T$ be a compact (metric) space, and let $\mathcal{X}$ be a Banach space over $\mathbb{K}$. Let $f : T \longrightarrow \mathcal{X}$ be a continuous ...
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Separation of two disjoint convex closed sets

Assume that $A,B\subset \mathbb{R}^n$ are two disjoint closed convex sets. Without using that $A$ and $B$ are closed sets, it follows already, that there is a non zero element $v$ and a real number $c$...
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Uniqueness of Hahn-Banach extensions for $c_0 \subseteq \ell^\infty$

Let $c_0$ be the space of real sequences converging to zero with supremum norm. $c_0$ is a (closed) subspace of $\ell^\infty$, the space of bounded real sequences. A $f \in {c_0}^*$ corresponds to a $...
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Confusion about Hahn-Banach

I am confused about something related to Hahn-Banach. According to my book, one corollary of H-B is that for $X$ a real or complex normed space, there exists $f \in X'$ such that $\|f\| = 1$ and $f(x) ...
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36 views

Hahn-Banach and the Fundamental Theorem of Calculus for Banach-space valued functions

I am trying to understand the proof of the Fundamental Theorem of Calculus for Banach space-valued functions, and in particular, how the Theorem of Hahn-Banach is being applied there. In the ...
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Find a norm in the dual space

Let $X$ be a normed space and $Y$ a linear subspace of $X$. We define $$Y^{\perp}=\{f\in X^*: f(y)=0, \; \forall y\in Y\}$$ and $$\|f\|_Y=\sup\{|f(y)|: y\in Y, \; \|y\|=1\}.$$ Prove that $$...
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Hahn Banach Theorem implying existence of a nonzero linear functional taking 0 in a linear subspace

I am reading this paper. In the proof of theorem 1, it is stated By the Hahn-Banach theorem, there is a bounded linear functional on $C(I_n)$, call it $L$, with the property that $L\ne 0$ but $L(R)...
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Proof explanation: Hahn-Banach Theorem in Stein's functional analysis

Theorem: Let $V$ be a linear space and $V_{0}$ is its subspace. Also let $p$ be a finite convex functional in $V$ and $l_{0}$ is linear functional in $V_{0}$. While $l_{0}(v)\le p(v),v\in V_{0}$ ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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, ...
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1answer
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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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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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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 ...
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1answer
91 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, ...
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2answers
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(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+...
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1answer
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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
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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 ...
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68 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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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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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 ...