# Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

4,162 questions
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### Sequences of Continuous Linear Operators between Banach Spaces.

Let $E, F$ be two Banach Spaces. Let $\{ T_{n} \}$ be a sequence of continuous linear operators from $E$ into $F$ such that: For all $x \in E: T_{n}x \rightarrow Tx,$ some limit in $F$. Then the ...
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### functional analysis : problem related to closed graph theorem

enter image description here the problem above is in Conway's [Functional Analysis] (p.93) it seems to be an application of closed graph theorem if the inequality were posed the other way it could ...
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### Uncountable basis in Hilbert space vs orthonormal basis

It is known that an infinite dimensional Banach space does not have a countable Hamel basis. It is also known that a separable Hilbert space has an orthonormal countable basis. Now, I think this basis ...
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### $A \in \mathcal{L}(X,Y) \implies A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$

Exercise : Let $X,Y$ be Banach spaces and $A \in \mathcal{L}(X,Y)$. Show that $A^* \in \mathcal{L}(Y^*_{w^*}, X^*_{w^*})$. Attempt : The linearity is trivial. Τo show that $A^*$ is $w^*$ to $w^*$...
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### Approximating a Banach space valued function by sums of continuous functions

I am trying to prove the following exercise, which is a part of a project type homework problem. Please give hints and suggestions, and discuss this problem. Let $(T,d)$ be a compact metric space, ...
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