Skip to main content
MathOverflow

Return to Question

Corrected typo in title.
Link
edit approved Jun 8, 2018 at 14:47
Community Bot
edit approved Jun 8, 2018 at 14:47
Community Bot
  • 1
  • 2
  • 3

Does seperabilityseparability of the strong operator topology imply separability of the underlying space?

added 6 characters in body; edited title
Source Link
Tomasz Kania
Tomasz Kania
  • 11.3k
  • 2
  • 39
  • 75

An equivalent condition for Does seperability of the strong operator topology imply separability of Banach spacesthe underlying space?

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.

AssumeSuppose that the strong operator topology on $B(X)$ is separable and cardinalthat the cardinal number of $B(X)$ is of the continuum.

Question. Can we conclude the norm topology on $X$ is separable?

Remark. It was proved by Tomek Kania that the assumption $|B(X)|=\mathfrak{c}$ is not solely enoughenough to get the separability assertion.

An equivalent condition for separability of Banach spaces

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.

Assume the strong operator topology on $B(X)$ is separable and cardinal number of $B(X)$ is of the continuum.

Question. Can we conclude the norm topology on $X$ is separable?

Remark. It was proved by Tomek Kania that the assumption $|B(X)|=\mathfrak{c}$ is not solely enough to get the separability assertion.

Does seperability of the strong operator topology imply separability of the underlying space?

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.

Question. Can we conclude the norm topology on $X$ is separable?

Remark. It was proved by Tomek Kania that the assumption $|B(X)|=\mathfrak{c}$ is not solely enough to get the separability assertion.

Source Link
ABB
ABB
  • 4.1k
  • 1
  • 11
  • 19

An equivalent condition for separability of Banach spaces

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.

Assume the strong operator topology on $B(X)$ is separable and cardinal number of $B(X)$ is of the continuum.

Question. Can we conclude the norm topology on $X$ is separable?

Remark. It was proved by Tomek Kania that the assumption $|B(X)|=\mathfrak{c}$ is not solely enough to get the separability assertion.