# 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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### Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
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### Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...
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### Is $\ell^1$ isomorphic to $L^1[0,1]$?

Can there be a continuous linear map, with a continuous inverse, from $l^{1}$ to $L^{1}(m)$ where $m$ is the Lebesgue measure on the unit interval $\left[0,1\right]?$ My thinking to this should be ...
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### One more predual space of the space of measures?

I am considering the Banach space $A$ with $\sup$-norm, which is the uniform closure of functions on a segment that are continuous but a finite collection of jump points, where they have limits from ...
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### Are nuclear Montel spaces projective?

The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
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### Urysohn-like theorem in Banach spaces

I have a (separable) Banach space $E$ and two closed disjoint sets $F$, $G$ in $E$. Now I wish to prove the existence of a $C^2$-function (Fréchet differentiable) $f:E \to \mathbf R$ that is $1$ on ...
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### Invertible bounded linear operators and closure

Let $T$ be an element of $B(X,X)$ (the bounded linear operators from X to itself), and let $W$ be a subset of $X$. Show that $T(\overline{W}) \subset \overline{T(W)]}$. Furthermore, if $T$ has bounded ...
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### Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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### Proof of the Goldstine theorem

I have a question concerning the proof of the Goldstine theorem, that the image of the unit ball $B_X$ of some Banach space $X$ is dense in the unit ball $B_{X^{**}}$ of its bidual under the canonical ...
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### Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...