Under what condition on the space X, any Continuous operator will be Completely continuous.

Question

Categorise the spaces $X$ for which $B_{00}(X,X)=B(X,X)$, where $B(X,X)$ is the set of bounded linear operators and $ B_{00}(X,X)$ the set of completely continuous operators, i.e. operators which take weak convergent sequences to strong convergence sequence.

Answer 1

This is equivalent to $X$ having the Schur property (i.e., each weakly convergent sequence converges strongly). A prominent non-finite-dimensional example is $X=l_1$.

If $B(X)=B_{00}(X)$ then $\operatorname{Id}\in B_{00}(X)$ and $X$ has the Schur property.

Let $X$ have the Schur property then $B(X)=B_{00}(X)$ trivially.