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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Let $X$ be normed vector space over $K$.

Let $\map \LL {X, X}$ denote the set of all linear transformations from $X$ to itself.

Let $\map C {X, X}$ denote the continuous mapping space from $X$ to itself.

Suppose:

$\map {CL} X := \map {CL} {X, X}$

Since $\struct {\map {CL} X, *}$ is an associative algebra, by definition of the continuous linear transformation space from $X$ to itself, $\map {CL} {X, X}$, we have:

$\map {CL} {X, X} := \map C {X, X} \cap \map \LL {X, X}$

By Identity Mapping is Continuous, we have that $I \in \map C {X, X}$.

By Identity Mapping on Normed Vector Space is Bounded Linear Operator, we have that $I \in \map \LL {X, X}$.

Hence $I \in \map {CL} X $ by definition of identity element.

Sources

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