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