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Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said abo …

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 …

[I have posted this question on MSE some time ago, but received no answer.] The title basically says all of it. If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I w …

Originally asked on MSE. Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ end …

Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think …

Jochen Wengenroth suggested to look at Carleson's interpolation theorem, and it seems like it completely answers my question. Namely, the following is true. Let $E$ be a subset of $D$. Then $H^\infty …

The following is essentially a partial case for my previous question. Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$ …

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able …

Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with …

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a …

Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed subspa …

I'll assume that $X$ is Hausdorff. A Hausdorff topological vector space is metrizable iff it is first countable. Hence, $X$ is metrizable and has a dense subset $\{x_n\}_{n=1}^{\infty}$. Define $f_n( …

Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that $P\left|_{X} …

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or comp …

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let …