Continuous Linear Transformation Algebra has Two-Sided Identity

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Theorem

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

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

Let $\struct {\map {CL} X, *}$ be an associative algebra.

Then there exists an identity element $I \in \map {CL} X$ such that:

$\forall x \in X : \map I x = x$

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