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)