# 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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### defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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### If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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### Understanding Lang's Proof of Fubini's Theorem

This question concerns the proof of Theorem 8.4 (Fubini's Theorem part 1) on page 162 in Lang's real and functional analysis book. To understand the proof I need to give following background from the ...
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### Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
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### Stable local minimizers of functions on a Banach space

Let $X$ be a Banach space and $f:X\rightarrow (-\infty,\infty]$ be a lower semicontinuous function. We are interested in some conceptions of local minimizer: We say that $\bar{x}\in X$ is a stable ...
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### Density of Banach spaces

I am trying to understand the notion of density in the context of Banach spaces. The density of a topological space is the least cardinality of a dense subset. Thus, a separable Banach space has ...
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### Dominated convergence theorem for Banach space

I'm trying to prove dominated convergence theorem for Banach space. Could you verify if my proof is fine or contains some subtle mistakes? Let $(f_n)$ be a sequence in $\mathcal L_0 (X, \mu, E)$. ...
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### $j(X)$ weak*-dense in $X^{**}$, $j$ is the canonical embedding

Let $X$ be a Banach space and consider the canonical embedding in it's bidual $X^{**}$, namely $j:X\to X^{**}, \; x\mapsto j(x)$, where $j(x)(x^*)=x^*(x)$ for $x^*\in X^*$. My question: Why is $j(X)$...
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### renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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### Complemented Banach spaces.

Let $X$ be Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties: $Z\cap Y=\{0\}$ and every $x\in X$ can be written in a unique ...
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### Spectrum of restriction to invariant subspace

Let $H$ be a separable Hilbert space and let $\mathcal{B}(H)$ denote the algebra of linear bounded operators on $H$. Let $T \in \mathcal{B}(H)$ and let $M$ be a non-trivial closed invariant subspace ...
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### Dual Space of General $L^p$ Space Which Takes Values in Banach Space

Last week, our functional analysis course covered Riesz Representation theorem for $L^p(X,\mu),(1\leq p < \infty)$, namely, $(L^p(X,\mu))^* = L^q(X,\mu)$. And I was stuck with this homework problem ...
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### How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
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### Is the space of $C^k$ submanifolds a Banach manifold

Let $M$ and $N$ be (finite-dimensional) smooth manifolds without boundary. For simplicity, assume $M$ is compact. This post concerns spaces of "copies of $M$ in $N$". To the best of my knowledge, the ...
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### Stability of point spectrum

Suppose $T$, $S$ are bounded operators on $l_2$, $a_n\to 0$ a sequence of complex numbers with the property that for any $n\in\mathbb{N}$, $T+a_nS$ has discrete spectrum and non-empty point spectrum....
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### Areas of research / interest in Banach space theory today.

Whilst taking a class in functional analysis I couldn't help but feel Banach space theory was only ever taught as a natural stepping stone towards Hilbert space theory. As to the prominence and status ...
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### Property of vector-valued measure

Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation. In general, if $B$ doesn't have the Radon-...
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### Banach space is separable iff Borel $\sigma$-algebra coincides with $\sigma$-algebra generated by open balls
The question is essentially in the title. Let $X$ be a Banach space, $\mathcal B(X)$ be a Borel $\sigma$-algebra, $\Sigma$ be a $\sigma$-algebra generated by the collection of all open balls. Is it ...