# 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.

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### Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
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### Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
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### Strong and weak convergence in $\ell^1$

Let $\ell^1$ be the space of absolutely summable real or complex sequences. Let us say that a sequence $(x_1, x_2, \ldots)$ of vectors in $\ell^1$ converges weakly to $x \in \ell^1$ if for every ...
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### A Banach space is reflexive if and only if its dual is reflexive

How to show that a Banach space $X$ is reflexive if and only if its dual $X'$ is reflexive?
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### Inequality between $\ell^p$-norms

Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$). Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
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### Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
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### $C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $$\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx.$$ I have to ...
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### How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
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### How do I prove the completeness of $\ell^p$?

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ $\Rightarrow$...
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### Weak-to-weak continuous operator which is not norm-continuous

Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is a) weak-to-weak continuous b) weak*-to-weak* continuous ($X=Y^*$) but not norm-to-norm ...
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### A Hamel basis for $\ell^p$?

I am looking for an explicit example for a Hamel basis for $\ell^{p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
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### At most finitely many (Hamel) coordinate functionals are continuous - different proof

If $X$ is a vector space over $\mathbb R$ and $B=\{x_i; i\in I\}$ is a Hamel basis for $X$, then for each $i\in I$ we have a linear functional $a_i(x)$ which assigns to $x$ the $i$-th coordinate, i.e....
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### How slow/fast can $L^p$ norm grow?

This is actually an exercise in Rudin's Real and Complex Analysis, $L^p$ spaces chapter. Could anyone help me out? Thanks in advance. Motivation: It's well known that if we have a function $f$ which ...
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### Is the weak topology sequential on some infinite-dimensional Banach space?

Recall that a topological space is sequential, iff every sequentially closed set is already closed. Is there an infinite-dimensional Banach space on which the weak topology is sequential? I ...
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$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
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### Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help. (By the way- this question does come from home-work, but I've ...
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### Example of different topologies with same convergent sequences

It's well known that for metric spaces the following is true Let $X$ be a space with two different metrics $d_1,d_2$ such that the two topological spaces $(X,d_1),(X,d_2)$ have the same ...
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### Proof of Pitt's theorem

I'm reading the book Topics in Banach Space Theory by Albiac F. Kalton N. J. I got stuck at the proof of Pitt's theorem. In the second paragraphs authors tries to prove ad absurdum that for weakly ...
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### If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)

Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But Wikipedia writes that ...