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We will show that if $F$ is a Banach space, and $\{f_n\}_{n\in\mathbb{N}}$ is a weak basis, it is a basis. For $n\in\mathbb{N}$ define $P_{n}$ on $F$ by $P_{n}f=\sum\limits_{k=1}^{n}a_{k}e_{k}$, where …

Let $D$ be a unit disc (I am actually interested in a much more general setting, but let's start with explicit examples). Let $E$ be an open subset of $D$. Consider the functional on the space of all …

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform con …

In case, anyone is interested, my question happened to be rather simple. Set-open topology coincides with the uniform topology if the target space is homogeneous enough. More precisely: Let $X$ be a …

Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space. Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B} …

[I have posted this question on MSE some time ago, but received no answer.] The title basically says all of it. If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I w …

Let $B\subset D$ be dense and countable. Let $B'\subset A^{2}(D,e^{-\varphi})^*$ be the set of point evaluations on $A^{2}(D,e^{-\varphi})$ at the points of $B$. Any $f\in A^{2}(D,e^{-\varphi})$ is co …

Let $f$ be an element of the Hardy space $H^2$, i.e. $f$ is an analytic function on the unit disk such that $\sum|a_n|^2<\infty$, where $f(z)=\sum a_n z^n$. Assume also that $f\not\equiv 0$. Is th …

First, let us assume that $f_2$ is continuous, and let $h=f_1-f_2$, which is an upper semi-continuous affine function, negative on the extreme points of $K$. Let $c=\sup_K h$. Since an upper semi-con …

Attempt number 2. Consider the case $f\ge 0$. For $\alpha\in[0,1]$ let $A_{\alpha}=\{x\in X, f(x)\ge \alpha\}$, which is measurable. For $n\in \mathbb{N}$ define $f_n=\frac{1}{n}\sum_{k=1}^{n}\chi_{ …

Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that $P\left|_{X} …

Here is an explicit answer. First, define a continuous tent map $w$ on $[1,2]$ define $w$ with $w(1)=w(2)=0$, and $w(3/2)=4$. Then define $w:[0,+\infty)\to [0,+\infty)$ by pasting scaled versions of t …

Originally asked on MSE. Let $E$ be a metrizable locally convex topological vector space and let $E^{*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (close …

Originally asked on MSE. Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ end …

Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said abo …