# Questions tagged [separable-spaces]

For questions about separable spaces, i.e., topological spaces containing a countable dense set.

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### The separability of $\mathbb{N}$ vs. the metrizability of $\ell ^1$.

Good afternoon, currently I'm trying to find an intuitive example for a space $E$, which is not separable, such that the space of finite measures $\mathcal{M}_f (E)$ on it is not metrizable. In ...
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### The space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable

How can one prove that the space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable? I am trying to think what can contradict seperability, but I didn't have any progress.
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### Relating to a topology and its basis.

Exercise. Show that for the topology $\tau$ in $\mathbb{R}$ with the subbasis sets $A = \{y \in \mathbb{R} : x \leqslant y\}$ and $\mathbb{R} \backslash A = \{y \in \mathbb{R} : x \not\leqslant y\}$as ...
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### $Z$ normed space : $\forall X\in SB\exists Y\subset Z : X,Y$ isomorphic. Show $\exists c>0 \forall X \in SB \exists Y \subset Z : X ,Y$ c-isomorphic.

I am currently studying a functional analysis course as part III of the Cambridge Tripos. In the course we were quoted the result: Let $SB$ be the class of separable Banach spaces over $\mathbb{R}$. ...
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if someone could give me an idea of ​​how to carry out this demonstration I would greatly appreciate it, let E be a Banach space, demonstrate E is separables $\Rightarrow$ The closed unit ball $B_E=\{... • 15 0 votes 1 answer 49 views ### Separable Banach space, unit ball, unit sphere demonstration [closed] hello I have the following question, about a Banach space E, prove they are equivalent: 1.E is separables 2.The closed unit ball$B_{E}=\{x\in E :||x||\leq 1 \}$is separables The closed unit ball$...
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Let $X$ be a linear space and let $\|.\|_1$, $\|.\|_2$ be two norms on $X$ such that $X_i:=(X,\|.\|_i)$ is separable for $i=1,2$. Then also $Y:=(X,\|.\|_1+\|.\|_2)$ is separable. I tried the following:...