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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 comp …

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ conver …

It seems that I have found a counter example myself. For the Hilbert matrix $$ H_\lambda:= \big( \frac{1}{1-\lambda+k+n} \big)_{k,n\geq 0}, \lambda < 1 $$ Rosenblum in "On the Hilbert Matrix I, Pro …

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all seque …

The opposite inequality cannot be true. If that were true, then consider a positive function $g$ with the property such that for all $s\in \mathbb{T}$ it holds that $g(s,x) \leq C g(s,y)$ whenever $|x …

So first of all, what is claimed in Love's paper is slightly different, It says that for a sufficient large choice of the parameter $c>0$ the function $$ g(x): = \sum_{n=0}^\infty c^{-n/p}\cos(c^n \pi …

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $$ \su …