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.
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 $L(X).$