Space of Linear Operators

Let and be Normed Spaces.
Write (or ) for the set of continuous Linear Operators
equipped with the Operator Norm .

Lemma

The space is a Normed Space.

Proof

Note that if then .
Indeed
so is bounded,
so it is a Linear Operator

Proposition

Suppose are Normed Spaces.
Let and .
Then and

Properties

Closed subspace of L(X,Y)
Completeness of L(X,Y)