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The following problem was left as an exercise in my lecture of Functional Analysis course and I am not able to solve 2 parts of them.

Show that the spaces $c_0(\mathbb{N}) $ is separable and $l^p(\mathbb{N})$, when $1\leq p < \infty$ is separable and not separable when p equals infinity.

Also, if $X= c_0(\mathbb{N})$ and $l^p(\mathbb{N})$ and $1 \leq p \leq \infty$, then show that $L(X)$ is not separable.

Now, I have managed to show that for $1\leq p < \infty$ , spaces $c_0(\mathbb{N}) $ and $l^p(\mathbb{N})$ are separable by constructing countable dense sets whose closure is X.

I have also managed to show that when $p= \infty$ , then $l^p(\mathbb{N})$ is not separable by constructing counter example.

( I am not adding the proofs as they were really tedious and will take a lot of time to write).

But I am not able to make reasonable progress in the case of $L(X)$ and in any of the spaces.

Can you please help me with the constructing counterexample in the case of any one of the space for L(X)?

I am badly struck on this problem for the $L(X)$ case.

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For any of these spaces $X,$ the space $\ell^\infty(\Bbb N)$ embeds in $L(X),$ the image of a bounded sequence $a=(a_n)\in\ell^\infty(\Bbb N)$ being the multiplication operator $T_a\in L(X)$ defined by $$\forall b\in X\quad (T_a(b))_n=a_nb_n. $$ (One easily checks that the operator norm $\|T_a\|$ is equal to $\|a\|_\infty$.)

Since $\ell^\infty(\Bbb N)$ is not separable, neither is $L(X).$

Anne Bauval
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  • @AnneBauvel "the image of a bounded sequence a=(a_n)". Image under what function? –  Dec 31 '22 at 14:55
  • This is the definition of the function! (the embedding), from $\ell^\infty(\Bbb N)$ into $L(X).$ Call it $T$ if you like, since I called $T_a$ the image of $a.$ @Isabella (Bauval, not Bauvel, but "happily", I discovered your message when being notified that you un-accepted my answer.) – Anne Bauval Dec 31 '22 at 15:23
  • @AnneBauvel I will soon accept it again. Very sorry for the typo in your last name. –  Dec 31 '22 at 16:58