Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \c …

I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with …

I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$". …

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when c …

Let $H^2(\mathbb{R}^3)$ the usual Sobolev space and consider the following set $$X=\bigg\lbrace u\bigg |u=\phi+\frac{Q}{|x|},\phi\in H^2, Q\in\mathbb{C}\bigg\rbrace$$ I observe tha …

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach …

Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is: 1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are ther …

Given a compact subset $A$ of a Banach lattice $E$, is the following true? There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$. This is true in case $E=C(X)$, $X$ c …

Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X. $\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphis …

Does anyone knows whether the set of the absolutely functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel se …

Let $X$ and $Y$ be separable Banach spaces and $L(X,Y) $ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace …

Let $B(H)$ is the C*-algebra of all bounded linear operators on Hilbert space $H$. Are there a closed two-sided ideal $I$ and a subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus …

Reading some parts of Benyamini-Lindenstrauss "Geometric non-linear functional analysis", I got curious about the fact that, in uniformly convex Banach spaces, there exists nonline …

Let $B_p := \{ x \in \mathbb{R}^d:\; \|x\|_p \le 1\}$ where $\|x\|_p := (\sum_{i=1}^d |x_i|^p)^{1/p}$ is the $\ell_p$ norm. (1) Let $t \in (0,1)$. Can we give an estimate on $$\m …

Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotie …