Continuous Linear Transformation Algebra has Two-Sided Identity

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:


 * $\forall x \in X : \map I x = x$