Analysis - Limit of the Average of the first N terms of a Sequence

Proving in Real Analysis Course. Please see the attached.

Banach Space Isomorphism Prove that “ ”, that is, prove that there is a Banach space isomorphism [Hint: for each in , define for each in Show that and .]

Functional Analysis Banach space Prove that “ ”, that is, prove that there is a Banach space isomorphism [Hint: for each in , define for each in Show that and .] *See attachment for complete question, inclu... click for more

Bounded Linear Operator; Bounded Invertible

Please solve the attached problems on bounded linear operators and bounded invertible equations.

Hahn Banach Theorem Application

Please assist me with the attached Hahn Banach Theorem problem. Thanks very much.

E is Nomed Linear Space ... Hahn-Banach Theorem.

Suppose that E is a normed linear space. Prove that if E* is separable, then E is separable. **See attachment for complete problem. Thanks!