1. ## separability problem

Hello! At the moment I'm preparing for an exam, and I'm stuck trying to answear the following (any suggestions are welcome):

Let H be a Hilbert space and denote by L(H) the space of all continuous linear maps from H to H (L(H) a Banach space). Suppose that the dimension of our H is infinite. Is L(H) separable? Why?

I'm thinking it's not. However, I've got nothing but intuition to back that up with.

2. ## Re: separability problem

Originally Posted by kenza
Hello! At the moment I'm preparing for an exam, and I'm stuck trying to answear the following (any suggestions are welcome):

Let H be a Hilbert space and denote by L(H) the space of all continuous linear maps from H to H (L(H) a Banach space). Suppose that the dimension of our H is infinite. Is L(H) separable? Why?

I'm thinking it's not. However, I've got nothing but intuition to back that up with.
Think of the elements of L(H) as matrices (with respect to some orthonormal basis). The diagonal matrices correspond to bounded sequences. The Banach space $\ell^\infty$ of bounded sequences is nonseparable ...

3. ## Re: separability problem

Oh! I see. Is that the same as saying that I found something "within" my L(H) that is not separable and L(H) is therefore not separable?

4. ## Re: separability problem

Originally Posted by kenza
Oh! I see. Is that the same as saying that I found something "within" my L(H) that is not separable and L(H) is therefore not separable?
Exercise: Every subspace of a separable metric space is separable.