### Prove the following proposition. Proposition ( 0.1 ) Let ( X ) and ( Y ) be reflexive Banach spaces. Assume that ( X ) is compactly embedded into ( X_{0} ), i.e., ( X subset X_{0} ) and every bounded sequence in ( X ) has a sub-sequence converging strongly in the norm of ( X_{0} ). Let ( T ) be a bounded linear operator from ( X ) to (

Prove the following proposition. Proposition ( 0.1 ) Let ( X ) and ( Y ) be reflexive Banach spaces. Assume that ( X ) is compactly embedded into ( X_{0} ), i.e., ( X subset X_{0} ) and every bounded sequence in ( X ) has a sub-sequence converging strongly in the norm of ( X_{0} ). Let ( T ) be a bounded linear operator from ( X ) to (