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Results tagged with fa.functional-analysis
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user 54609
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
24
votes
Accepted
Are proper linear subspaces of Banach spaces always meager?
I am afraid that Konstantin's accepted answer is seriously flawed.
In fact, what seems to be proved in his answer is that $\ker f$ is of second category, whenever $f$ is a discontinuous linear functi …
- 716
4
votes
Accepted
Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells
Let $x_n=-\frac{\sum_{k=1}^n e_k}{n}$. Then $\|e_i-x_n\|^2=\frac{n+3}{n}$ if $i \leq n$ and $\|e_i-x_n\|^2=\frac{n+1}{n}$ if $i>n$. So your set is the intersection of the sets $B(x_n,\sqrt{\frac{n+1}{ …
- 716
24
votes
Accepted
Are uniformly continuous functions dense in all continuous functions?
Yes, and even more is true. The argument is as follows: let $f\colon X \to \mathbb R$ be a continuous function and let $K\subset X$ be a compact set. Then $f|_K$ is uniformly continuous; let $\omega$ …
- 716
4
votes
relation between of uniformly rotund in every direction and uniformly rotund and locally uni...
First, the fact that UR implies URED is obvious (the definition of UR requires less from the sequences $(x_n)$, $(y_n)$ in order to conclude the convergence $\|x_n-y_n\|\to 0$.)
Second, the notions L …
- 716