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Let $X$ and $Y$ are Banach spaces and $B(X, Y)$ be the Banach space of bounded linear operators from $X$ to $Y$ with operator norm.

What are the conditions under which $B(X, Y)$ is separable?

Jakobian
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PAMG
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    In the case of Hilbert spaces $B(X,Y)$ is separable iff $X$ is finite dimensional and $Y$ is separable or $Y$ is finite dimensional and $X$ is separable. – Kavi Rama Murthy Aug 26 '19 at 10:29
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    As Robert Israel pointed out here there exist infinite-dimensional separable Banach spaces $X,Y$ such that $\mathcal B(X,Y)$ is norm-separable. Generally speaking---at least this is the feeling I have---the situation for arbitrary Banach spaces seems to be very messy. – Frederik vom Ende Jan 02 '20 at 18:21
  • One thing to note is that if $X = 0$ or $Y = 0$ then $B(X, Y) = 0$ is always separable. And if $B(X, Y)$ is separable with $X, Y\neq 0$ then $B(X, Y)$ contains an isometric copy of $X^*$ and there exists a continuous surjection of $B(X, Y)$ onto $Y$, so that both are separable. – Jakobian Oct 02 '24 at 15:55
  • I believe that if $X, Y\neq 0$ and $X$ is finite-dimensional, then $B(X, Y)$ is separable iff $Y$ is separable, and $Y$ is finite-dimensional, then $B(X, Y)$ is separable iff $X^$ is separable. So that focus of study should be on when $X, Y$ are both infinite-dimensional with $X^$ and $Y$ separable (this also implies $X$ is separable). – Jakobian Oct 03 '24 at 03:29

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