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Let $(X,d)$ be an uncountable infinite complete disconnected metric space (what I have in mind is something like $X=\{0,1,\ldots,n\}^{\mathbb{N}}$). I would like to know if the space $C^{\gamma}(X)$ o …

Let $X$ be a locally compact Hausdorff space, $\beta X$ its Stone-Cech compactification and $\Delta: X\to\beta X$ the inclusion map. Given a Borel probability measure $\mu^{\beta}$ over $\beta X$, is …

I am working with transfer operators and I reach a point where would be nice if I could use a result from Tosio Kato's book about perturbation theory of linear operators. I think I am able to use Kato …

Consider an enumeration $\{q_1,q_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e_1,e_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define $Ae_{2k-1}=e_{2k-1}$ and $Ae_{2k}=q_ke …

Let be $(a_n)\in\ell^2(\mathbb N)$ and consider the mapping $f:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ given by $$ f\Big((a_n)\Big)=(a_n^n). $$ Question: Is $f$ a Fréchet $C^{\infty}$ function in whole …

Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose …

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$. Let be $g:[ …

Probably is not general as you want, but if you don't think before about that can be a begining... Proposition: If $\pi:Y\to X$ is bijective function such that $\pi^{-1}$ is continuous then $\mathbb …

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^ …