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Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is i …

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\in …

We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, …

There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\l …

Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with …

Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$. For which $X$'s is it true that $J_A+J_B=J_{A\cap …

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ vani …

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 …

I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling al …

I am looking for a paper "Linear spaces with disjoint elements and their conversion into vector lattices" by A. I. Veksler. It was published in 1967 in Research Notes of Leningrad State Pedagogical Un …

I think I've found a counterexample in the literature. I would really appreciate if somebody verified that I didn't get confused about the terminology of Orlicz spaces. Recall that a Banach space $E$ …

Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (note th …

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 $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map). Let $\sim$ be the minimal closed equivalence relation on $X$ tha …

Recall that a topological space is called Frechet-Urysohn if the operations of closure and sequential closure coincide. Let $X$ be a locally compact Hausdorff space. It is known that $C(X)$ is not Fre …