# Help me tie up loose ends in operator theory

Homework Helper

## Main Question or Discussion Point

Dear knowledgeable person,

please, help me with the following problems:

Let $A$ be a closed operator in the Banach space $(B, ||.||)$.
Let D(A) be its domain in B. Prove that D(A) endowed with the graph norm is a Banach space.

The graph norm is defined as:

let 'a' be a vector in D(A). Then $||a||_{\mathbb{graph}} := ||a||+ ||A a||$.

I can easily show that the graph norm makes D(A) a pre-Banach space, but what about completeness wrt the graph norm ? I'm sure it has to do with the closedness of A, but how ?

Also, how does one prove the quivalence of the 2 known definitions of a closed operator in a banach space ? Is it really trivial, as everyone claims ?

http://planetmath.org/?method=l2h&from=objects&name=ClosedOperator&op=getobj

Last edited: