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Questions tagged [hahn-banach-theorem]

Questions regarding the Hahn-Banach Theorem.

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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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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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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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(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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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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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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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 ...