# Tagged Questions

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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### there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
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### a compact operator on $l^2$ defined by an infinite matrix

Let $A$ be an infinite matrix such that $\displaystyle \sum_{i,j}|a_{i,j}|^2<\infty$. Then $A$ defined a compact operator on $l^2$.
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### $\mathcal M(K)$ is an $\mathcal{l}_1-$sum of $L_1(\mu)$ spaces

Let $K$ be a compact Hausdorff space. I want to show $\mathcal M(K)$ is an $\mathcal{l}_1$-sum of $L_1(\mu)$ spaces, where $\mathcal M(K)$ is the dual of $C(K)$. I have got the sketch of the proof but ...
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### Stone's Theorem Integral: Avanced Integral

Reference This problem grew out from: Stone's Theorem Integral: Basic Integral Problem Given the real line as measure space $\mathbb{R}$ and a Hilbert space $\mathcal{H}$. Consider a strongly ...
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### Subspace isomorphic to a complemented subspace

I'll begin by writing down the definitions I'm using, to avoid confusion. Let $X$ be a Banach space and let $Y$ be a subspace of $X$. We say that $Y$ is complemented in $X$ if there exists a linear ...
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### Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$

Seems simple enough but I can't quite get it. $X$ is a complex Banach space, and $T\in B(X)$, $T'\in B(X')$ is its adjoint. Suppose $T'$ is invertible. How can we show that $T$ is invertible? I have ...
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### Derivative Bilinear map

I wanted to calculate the derivative of a continuous bilinear map $B: X_1 \times X_2 \rightarrow Y$. (Does anyhere know whether there is a generalisation of the notation $L(X,Y)$ that you use for the ...
3answers
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### Spectrum of a nilpotent operator

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?
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### Is the complement of an open ball in a Banach space connected?

Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected? Idea One could ...
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### A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
4answers
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### Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
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### Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
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### What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...