# 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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### $h$ is an affine function?

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
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### quick question about renorming quasi-Banach spaces into p-Banach spaces

I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) ...
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### Elementary relationships between finite-dimensional subspaces and finite-codimensional subspaces

I have an elementary question about finite-dimensional subspaces and finite-codimensional subspaces. This question may be known. Question. Let $U$ be a finite-dimensional subspace of an infinite-...
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### Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
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### Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu )$ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$. If $X$ is reflexive, ...
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### The bidual of the space of divergence-free vector fields

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
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### complemented subspace of the direct sum of two Banach spaces

When I was reading a paper, I saw something like: If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
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### Closed graph theorem for cones?

In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are ...
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### Injective continuous operators between Banach spaces

Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty?
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### Isometries on the unit sphere

Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
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### Containment of $c_0$ in projective tensor products

Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product. Question: If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
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### Reference request: completion of Banach norm on sum

Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset $$X_1+X_2 = \left\{ x\in X:\, x= \beta_1 x_1 + \beta_2 x_2 \, \beta_i \in \mathbb{R},\, x_i \in X_i \right\},$$ a is ...
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### Density and the projective tensor product

Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set  D^+\...
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### Law of a step function and its generalization to two dimensions on an appropriate spaces

Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively: A step function: $u_1(x)=\begin{cases} u_{L}, x<c_1, \\[2ex] u_{R}, x>c_1, \end{cases}$ A "...
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### Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space: Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
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### Example of a nonconvex Chebyshev set in a metric space with continuous projection?

Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous? For convexity to be well-defined, we need to assume that $X$ is a vector ...
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### Banach lattices $X$ for which $L_p(\mu)\subset X$ or $X\subset L_q(\mu)$

It is well known (see vol. II of Lindenstrauss and Tzafriri's book) that an order continuous Banach lattice $X$ with a weak unit admits a representation as a (in general not closed) ideal of $L_1(\mu)$...
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### A closed subspace generated by open $F_{\sigma}$ sets of $K$ in $C(K)^{**}$

Let $K$ be a compact Hausdorff space. For a bounded Borel measurable function $f$ on $K$, we define $\phi_{f}\in C(K)^{**}$ by $\phi_{f}(\mu)=\int_{K}fd\mu$ for all $\mu\in C(K)^{*}$. It is easy to ...
Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ ...