Questions tagged [hahn-banach-theorem]

Questions regarding the Hahn-Banach Theorem.

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The dual of a subspace of a normed space $E$ is isometric to a subspace of $E'$

Let $E$ be a normed vector space and $F\subset E$ be a subspace of $E$. I want to show that there's a bounded linear operator $T:F'\to E'$ such that given $f\in F'$, $T(f)$ is an extension of $f$ and $...
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taking the orthogonal complement twice in a banach space

a direct consequence of the hahn banach theorem is the if $L\subset X$ is a closed subspace, then $(L^{\perp})_{\perp}=L$. what I'm trying to understand is whether or not for every $L\subset X^*$ ...
Aviv's user avatar
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2 answers
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Using Hahn Banach for switching between $L^p$ spaces

I want to understand the proof (or under which conditions a proof holds) of the following statement: Let $f$ be a function in $L^p$ and let $q$ be such that $\frac{1}{p}+\frac{1}{q}=1$. Then we can ...
proofromthebook's user avatar
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Strict form of Hahn Banach inequality

Let $B$ be a Banach space. Let $x\in B.$ Then by Hahn-Banach there exists $f\in B^\ast$ such that $\|f\| =1$ and $f(x) = \|x\|.$ Question: can we strengthen this result so that now, for each $\epsilon&...
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Confusion regarding statement of the Hahn-Banach theorem

I read about Hahn-Banach theorem from the book "Introduction to topology and modern analysis" by Simmons. There it was stated like this: Let $M$ be a linear subspace of a normed linear space ...
Anindita Sarkar's user avatar
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Strict separation (in a very weak sense) of two closed convex sets

Given $A,B\subseteq\mathbb R^n$ two disjoint, nonempty, closed, convex sets, can we always find a vector $v\in\mathbb R^n\setminus\{0\}$ such that $\langle v,a\rangle<\langle v,b\rangle$ for all $a\...
Mizar's user avatar
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Proof of Hahn-Banach theorem from Compactness theorem of FOL

Since the Compactness theorem of FOL is equivalent to the ultrafilter Lemma, which implies Hahn-Banach, the implication is clear to me. I was more just wondering if there is a nice direct proof? I saw ...
Niko Gruben's user avatar
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Functional Analysis problem about bounded linear operations [duplicate]

Let X and Y be normed spaces non-empty. If every bounded linear operator $T:X \to Y$, non trivial, is surjective, show that dimY = 1. Can someone help me with that? The clue is to use the Hahn-Banach ...
Luca's user avatar
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Distance of a functional to a subspace

When learning about closed range theorem, I met a problem as follows: suppose $X$ is a Banach space and $E$ is a closed subspace of $X$. Denote $$ E^{\perp}=\{g\in X^*\ |\ g(x)=0,\quad\forall x\in E\}....
MakaBaka's user avatar
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Prove that a Banach space cannot be reflexive if some strict closed subspace of its dual space separates its points [closed]

Let $X$ be a Banach space and let $Z$ be a closed subspace of $X^*$ such that $Z\neq X^*$. Suppose $Z$ separates the points in $X$, that is, if $x \in X$ and $x^*(x) = 0$ $\forall x^* \in Z$ then $x = ...
14Lucas07's user avatar
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A doubt on applying Hahn-Banach to problems

I had a doubt on Hahn-Banach continuous extension theorem. Wikipedia says this: Hahn–Banach continuous extension theorem — Every continuous linear functional $f$ defined on a vector subspace $M$ of a ...
MathRookie2204's user avatar
7 votes
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140 views

Show that no arbitrage implies the extension property in $L^p$

Let $(\Omega,\mathcal F,P)$ be a probability space, and let $X:=L^p$ denote the normed space of (equivalence classes) of $p$-integrable real random variables on $(\Omega,\mathcal F,P)$, where $1\leq p&...
Alphie's user avatar
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Hahn-Banach separation theorem: show that $f(a)<\inf f(B)$ for all $a\in \text{int}(A)$

Hahn-Banach separation Theorem. Let $A,B$ be convex non-empty disjoint subsets of a real normed space $X$. Suppose also that $\text{int}(A)\neq \emptyset$ (non-empty interior). Then there exists a ...
Alphie's user avatar
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Basic question about the control one has for extending a function using Hahn-Banach

Let $A$ be a closed subspace of a Banach space, $V$. I want to show there is a functional $f'$ such that $A \subset \ker(f')$ and $f'(x) \ne 0$ for $x \in V-A$. I'm wondering what control we have when ...
Ty Perkins's user avatar
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Proving the Existence of a Linear Functional on $\ell^{\infty}$ Bounded by $\limsup$

Let $X=\ell^{\infty}$ and $p: X \rightarrow \mathbb{R}, p(x)=\limsup _{n \rightarrow \infty} x_n$. Is there a linear functional $f: X \rightarrow \mathbb{R}$ such that $$ -p(-x) \leq f(x) \leq p(x) ? $...
CanDoMajoringMath's user avatar
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Inequality of supremum

I am going trough a funtional analysis course and to prove the geometric form of the Hahn-Banach theorem we need to prove that $\sup_ { \ b \in B } f(b) < \sup_ { \ b \in B \\ \|x\| < \epsilon.}...
Pablo Borrego's user avatar
1 vote
1 answer
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What is the corresponding Hahn Banach Separation Theorem?

Here is a separation theorem in $\mathbb{R}^n$: Let $A$ and $B$ be nonempty disjoint convex sets in $\mathbb{R}^n$. Then there exists a nonzero linear functional $L$ on $\mathbb{R}^n$ such that $\inf ...
Ypbor's user avatar
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Question about uniqueness of a specific Hahn-Banach extension

Let $X = C([0,1])$, and $L = Span(t)$. Defining the functional $f(x):= \lambda$ when $x \in L$ is $\lambda t$. Clearly, $||f||_{L^*} = 1$. From Hahn-Banach, it can be extended to $F \in X^*$, with $||...
domingoac's user avatar
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Uniqueness of Hahn Banach in $C[0, 1]$

I'm very recent to functional analysis. My homework problem states: Let $x_0(t) \in C[0, 1]$ a fixed continuous function and $ L = span(x_0) $. Consider the functional in $L$ defined as: $$ f(x) := \...
C. Alcaino S.'s user avatar
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Can the condition of nonempty interior be removed in the hyperplane separation theorem?

Consider the following hyperplane separation theorem: Suppose $E$ is a proper convex subset in a normed vector space $\mathcal{X}$ and $\theta$ (the origin) is an interior point of $E$. And suppose $...
W.J's user avatar
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Showing that set of all Hahn-Banach extensions of a functional is closed and has empty interior but need not be compact.

Let $Y$ be a subspace of a normed linear space $X$ and $g\in Y'$. How do we show that the set of all Hahn-Banach extensions of $g$ to $X$ is a non-empty, convex, closed and bounded subset of $X'$ ...
Babai's user avatar
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3 answers
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A factor von Neumann algebra is a prime algebra.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ denotes the algebra of all bounded linear operators on $\mathcal{H}.$ Recall that a von Neumann algebra $\mathcal{U}\...
MOHD ASIM's user avatar
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1 answer
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The algebra of all bounded linear operators acting on a complex Banach space is a prime algebra.

Let $X$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B}(X)$ the algebra of all bounded linear operators on $X.$ I want to show that $\mathcal{B}(X)$ is a prime algebra. My ...
Akhter's user avatar
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2 votes
1 answer
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Finding a compact set to apply the Hahn-Banach separation theorem in a locally convex topological vector space

I am trying to justify how the Hahn-Banach theorem was applied in the proof below. It looks like the proof is using the case for locally convex space (because the inequalities are strict). that ...
some_math_guy's user avatar
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About Hahn-Banach Theorem assumption

Currently I am trying to understand the proof of the following lemma which appear in the book Additive combinatorics of Tao and Vu (Chapter 5, page 212, in the 2006 edition). Lemma 5.14 Let $A, B$ be ...
Brien Navarro's user avatar
3 votes
1 answer
151 views

How do details of the proof of the universal approximation theorem work?

So I have a proof of the universal approximation theorem and need to understand how it works. So this is the theorem: Let $ d \in \mathbb{N} $, let $K \subseteq \mathbb{R}^d $ be compact, and let $\...
Lopsio's user avatar
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0 votes
1 answer
89 views

Riesz Lemma from Hahn Banach Theorem

A note by Jacob Manaker provides a proof of the Riesz lemma via Hahn Banach. I am unable to follow a key construction in this proof. I note the proof down below for reproducibility, along with a ...
Siddharth Bhat's user avatar
3 votes
2 answers
137 views

Every complemented subspace has a non topological algebraic complement

I am trying to understand the topological direct sum in normed vector spaces, i.e. the algebraic sum of two subspaces where the projections (or equivalently one of them) are continuous. I ran into ...
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Bochner integrable function is a.e. non-negative, if Bochner integral is non-negative on all sets

I know that the Hahn-Banach Theorem, implies the following statement for Banach-space valued functions: Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite complete measure space, and $(E, |\cdot|_E)$ a (...
Ata Keskin's user avatar
2 votes
1 answer
153 views

Non-existence of Banach-Tarski in the plane from non-existence on the line

The following theorem is well known: There exists a isometry invariant finitely additve measure, measuring all subsets of $\mathbb{R}^d$ that extends the Lebesgue measure if and only if $d\le 2$ ...
Vivaan Daga's user avatar
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3 votes
0 answers
55 views

How to extend a functional using Hahn-Banach Theorem

Hahn Banach Theorem: Let $X$ be a normed Vector space and $U$ a subspace. For every linear continuous function $u':U \rightarrow \mathbb{K}$, there exists a continuous linear functional $x': X \...
wanymose's user avatar
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Hahn-Banach theorem proof

I read the Brezis functional analysis, differential equations and Sobolev spaces textbook. We extend $g(x):G\to\mathbb{R}$ to $f(x):E\to\mathbb{R}$ The idea of the proof is to construct a well-ordered ...
KeepKolmogorov's user avatar
1 vote
1 answer
86 views

An application of Hahn-Banach theorem on the left-shift operator on $l^\infty$

I think this is a fairly standard problem and has been asked about before. However I am mostly interested in getting help with how to close the argument that I have attempted below. Task: The idea is ...
kapython's user avatar
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Geometric Hahn-Banach theorem and linear operators [duplicate]

I've got this problem. $(X, ||\cdot||)$ is a normed vectorial space. Then, $l_0, l_1, ..., l_n$ are linear operators in $X^*$ (dual space) and for each $i\in \{0,1,...,n\}$, let $ker\:l_i = \{x\in X | ...
Tomas Rojas's user avatar
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0 answers
68 views

Hahn-Banach theorem on locally convex spaces

In their book, Reed & Simon state the theorem (Theorem V.3): Let $X$ be a locally convex space and let $Y \subset X$ be a subspace. Let $\ell: Y \rightarrow \mathbb{R}$ (or $\mathbb{C}$ if $X$ is ...
CBBAM's user avatar
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0 votes
1 answer
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On the existence of Green's functions by Peter Lax

I'm reading Peter Lax's paper On the existence of Green's functions. He showed that the regular part of Green's functions is continuous on the boundary. My question is : to have the normal derivative ...
Wayne's user avatar
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-2 votes
1 answer
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Using Hahn-Banach theorem show that $\sum\limits_{i = 1}^{n} \alpha_i \beta_i\gt \lambda \geq \left \|\sum\limits_{i = 1}^{n} \beta_i f_i \right \|.$ [closed]

Let $V$ be a real Banach space and $B$ be the closed unit ball in $V.$ Let $f_1, \cdots, f_n \in V^{\ast}$ be norm continuous linear functionals on $V$ and $\alpha_1, \cdots, \alpha_n \in \mathbb R$ ...
RKC's user avatar
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2 votes
2 answers
112 views

How is the Hahn-Banach theorem is applied here?

I am trying to understand how the Banach theorem is applied to get the following highlighted conclusions. The only way I could think of is the following: Since the dual $X^*$ is a normed space, if $x^...
XXX1010's user avatar
  • 127
3 votes
1 answer
219 views

Is Hahn-Banach equivalent to the ultrafilter lemma in ZF

I know that the ultrafilter lemma is weaker than the axiom of choice (in ZF) And that in order to prove Choice in ZF from the ultrafilter lemma we need the Krein-Milman theorem so $UF+KM=AC$ ...
El Ruño's user avatar
  • 736
3 votes
0 answers
36 views

Variant of Analytic form of Hahn-Banach

The HB theorem states that given a linear functional $f: G \to \mathbb{R}$ where $G$ is a subspace of a normed vector space $E$ and a function $p:E \to \mathbb{R}$ satisfying $p(x+y) \leq p(x)+p(y)$. ...
Alek Fröhlich's user avatar
1 vote
1 answer
38 views

If two elements are different there is a functional under where the image is different

I have the following exercise in functional analysis: Let E be a normed space and $x,y$ different vectors. Prove or disprove finding a counterexample that it exist a function $f \in E^*$ such that $f(...
user1072285's user avatar
4 votes
0 answers
122 views

Prime Ideal Theorem implies Hahn Banach Theorem

I am reading Jech's Axiom of Choice, and there is this exercise: chapter 2 Problem 19: Show that the Hahn-Banach Theorem follows from the Prime Ideal Theorem. I came up with a (possibly wrong) proof,...
mathlearner98's user avatar
2 votes
1 answer
39 views

Problem with uniqueness of expansion of functional [duplicate]

Let $M$ be closed subspace of Hilbert space $H$ and let $f$ be bounded linear functional on $M$. Prove that there is unique expansion $F$ from $f$ on whole $H$ which satisfies $||F||=||f||$. Prove ...
Broj 1's user avatar
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1 vote
0 answers
82 views

Application of Hahn Banach for normed spaces

Let $X$ be a normed space. I want to show that - using Hahn Banach - for every $x$ we have $$||x|| = \sup\{|f(x)| : f \in X^*, ||f|| \leq 1\}$$ One direction is easy: $$|f(x)| \leq ||f|| \cdot ||x|| \...
MyGanton's user avatar
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1 vote
1 answer
55 views

Looking for a linear functional with given properties

Let $X$ be a normed $\Bbb{R}$-vector space. Let $Y \subset X$ be a closed subspace and $x_0 \in X \backslash Y$. We define $$d :=\text{dist}(x_0, Y) = \inf_{y \in Y}\,\lVert x_0 − y \rVert$$ I would ...
yrual's user avatar
  • 510
3 votes
0 answers
142 views

Which Banach spaces admit medians?

In a metric space $X$, let $I(x,y)$ denote the metric interval between two points, i.e. $I(x,y):=\{z:d(z,x)+d(z,y)=d(x,y)\}$. Given a triplet $(x,y,z)$, we say that $w$ is a median of the triplet if $...
Pelota's user avatar
  • 548
4 votes
0 answers
168 views

Proof verification that the Hahn Banach theorem equivalent to existence of a finitely additive measure for boolean algebra over ZF

Exercise 2.6.19 of Jech's Axiom of choice asks to show that the Hahn Banach theorem is equivalent to the existence of a real valued measure for all Boolean algebras over $ZF$. This is a sketch at my ...
MIO's user avatar
  • 1,916
1 vote
1 answer
113 views

Banach limit and extended limit

The linear functional $x \mapsto \lim x$ on space $c$ (of all convergent sequences) has norm $1$ and so by Hahn-Banach theorem possesses extensions $L$, of norm $1$, defined on space $\ell_{\infty}$. ...
Air's user avatar
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1 vote
1 answer
240 views

How to prove that the Hahn-Banach extension in this case is unique

I am struck on a problem in my operator theory class so I am posting it here in the hope of getting some help. Let $H$ be a Hilbert space and let $E\subseteq H$ be a closed subspace. Show that if $\...
user avatar
4 votes
2 answers
204 views

Existence of Banach limits: Translation invariance

A positive functional $\Phi$ on $\ell^{\infty}$ is said to be a Banach limit if $\Phi(1,1,1,\ldots)=1$ and $\Phi\circ L=\Phi$ where $L$ is the left shift operator on $\ell^{\infty}$. Show that there ...
Guest's user avatar
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