Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach …

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. …

Background Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F …

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem. Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such …

For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not comp …

On the one hand, Wikipedia suggests that every distribution defines a Radon measure: http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions On the o …

Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial …

Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one …

I'm looking for a good reference (preferably a book) that discusses methods of solving volterra equations of the second kind. For those who wand to know why I'm interested in volt …

Let $K$ be the closed unit ball of some infinite dimensional Banach space, and let $H$ be an autohomeomorphism of $K$, having fixed points. Can $H/2$ be fixed point free ? Also, l …

It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle. What i …

The result stated in the title is thoroughly standard - or that's the impression I got. I seem to remember seeing it stated somewhere in a book I was reading in the library, and t …

Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\math …

Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem? It states: For any infinite dimensional Banach space $X$ there is a $q &gt …

A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is: Which Fréchet manifolds have a smooth partition of u …