Answer to Question #153574 in Functional Analysis for Ashweta Padhan

Question

B(T) is separable iff T is finite.prove it

Answer 1

By theorem: A Hilbert space H is separable if and only if H contains a countable

set which is dense in H.

In our case: if the set of all continuous linear transformations B(T) is finite, it contains countable set of linear combinations of elements with rational complex coefficients.