Continuous Linear Transformation Space as Algebra

Theorem

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.

Let $* : \map {CL} X \times \map {CL} X \mapsto \map {CL} X$ be a bilinear mapping.

Then $\struct {\map {CL} X, *}$ is an associative algebra.

Proof

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Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations