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Let $X$ be a topological vector space. Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{x_ …

Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded m …

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$. Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum. …

Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?

Let $X$ be a Banach space. Is the following implication valid? $$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$ The converse is clearly true, since the close …

Let $X$ be a separable topological vector space with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi_n:X\to X$ pointwise converging …

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$. Q. Can we concluded that $X$ is hereditery Li …

Let us consider the space of convergent sequences which is denoted by $c$. The space of all sequences $(x_n)\in c$ with $\lim x_n=0$ is also denoted by $c_0$. Clearly $c_0$ is a proper closed subspac …

Q1. Does there exist a separable Banach space $X$ satisfying in the following property? 1- $X^*$ is non separable. 2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ suc …

Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$. Conven …

Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow $$w-\lim T_i=T \Longleftrightarrow \ …

Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?

A Lindelöf space is a topological space in which every open cover has a countable subcover. Does there exists a Lindelöf locally convex space which is not second countable? I am also looking for a …

I am looking for a topological vector space $(X,\tau)$ enjoying the following conditions: 1- $(X,\tau)$ is not locally convex. 2- There exists a metric $d$ on $X$ and a sequence $\{X_n\}$ of subset …

Let $X$ be a separable Banach space. Is the strong operator topology metrizable on $B(X)$, the space of all bounded operators on $X$? SOT-$\lim T_i=0~$ if and only if $~\lim \|T_ix\|=0$ for every $x …