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# 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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### Differential of non-Euclidean distance function

For a fixed value of $k \in \mathbb{N}$, let $P^{k}[\mathbb{R}]$ be the real vector space of all real polynomials of degree $< k$. Fix $k$ distinct values $x_1 < \ldots < x_k \in \mathbb{R}$ ...
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### Compute the norm of a bilinear operator on the space of Lipschitz functions vanishing at $0$

Let $M=[-1,1]\subset ℝ$. A function $f: M → ℝ$ is a Lipschitz function if there exists a finite non-negative constant $C$ such that $|f(s)-f(t)| ⩽ C\|s-t\|$ for all $s, t ∈ M$. Denote the set of ...
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### Has every homogeneous Banach space on the torus an operator S_n (Katznelson on Harmonic Analysis)?

In his book about Harmonic Analysis Katznelson defines homogeneous Banach spaces on the 1-dim. torus: in short, such a space is a linear subspace $B$ of $L^1(\mathbb T)$ with a norm denoted $||f||_B$ (...
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### Weirdest properties of Banach spaces?

This is soft question, but I've recently have become interested in the odd properties of Banach spaces after studying Banach spaces which do not admit a pre-dual. I also almost work exclusively with ...
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### Completeness of Y (normed tvs) in the proof of th. 2 in ch. V section 3 of Yosida's Functional Analysis ed 6

I'm stuck on the proof of theorem 2 in chapter V, section 3 of Yosida's Functional Analysis edition 6 (pages 140,141). Theorem 2 says : A locally convex linear topological space X is reflexive iff it ...
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### Coordinate-wise projections separate points in the closed span of a sequence

Let $X$ be a Banach space over $\mathbb K$ where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $(x_n)$ a linearly independent sequence (can be normalized with necessary or convenient) in $X$, $E$ ...
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### Convex hull of cartesian product in general vector spaces

Is it true that $\text{conv}(X)\times\text{conv}(Y)\subset \text{conv}(X\times Y)$, where $X,Y$ are subsets of a (not-necessarily-finite-dimensional) vector space? If the answer is “no”, what if we ...
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### Can angles be defined by norms that are not induced by inner products?

Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
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### Prove the compactness theorem for Radon measures by using Banach-Alaoglu theorem

I was reading the proof of the compactness theorem for Radon measures (Theorem 1.5.15) from Leon Simon's book: Geometric Measure Theory I was confused by the highlighted part. I hadn't learned the ...
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### Split and differentiable locally trivial submersion, and more confusions on a proof in John H Hubbard's book

The "split" means Let E, F be Banach spaces, A surjective linear map $f:E\rightarrow F$ is said to split if $ker f$ admits a closed complement. More generally, a submersion is a split ...
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