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Dear knowledgeable person,

please, help me with the following problems:

Let [itex] A [/itex] be a closed operator in the Banach space [itex] (B, ||.||) [/itex].

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 [itex] ||a||_{\mathbb{graph}} := ||a||+ ||A a|| [/itex].

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 ?

See also here

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

please, help me with the following problems:

Let [itex] A [/itex] be a closed operator in the Banach space [itex] (B, ||.||) [/itex].

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 [itex] ||a||_{\mathbb{graph}} := ||a||+ ||A a|| [/itex].

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 ?

See also here

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

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