# functional analysis – For a compact operator \$T in K(X, H)\$ and H an Hilbert space, \$overline{T(X)}\$ is separable

Let X be a normed space, H be a Hilbert space, and let $$T in K(X, H)$$ the set of compact operators. Show that T(X) is separable. I tried to use the fact that the set of finite rank operators $$F(X,H) = K(X,H)$$ when H is an Hilbert space, to show that $$T(x)$$ is countable (since we need to show that $$overline{T(X)}$$ is separable, that is, that it contains a countable dense subset)