Continuous Linear Operator over Infinite Dimensional Vector Space is not necessarily Invertible

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 $I \in \map {CL} X$ be the identity element.

Let $S, T \in \map {CL} X$.

Suppose the dimension of $X$ is finite:


 * $d = \dim X = \infty$

Suppose $T \circ S = I$ where $\circ$ denotes the composition of mappings.

Then $T$ and $S$ are not necessarily invertible.

Proof
Let $X, Y$ be $2$-sequence spaces.

Let $T = L : X \to Y$ be the left shift operator.

Let $S = R : X \to Y$ be the right shift operator.

Let $x := \tuple {a_1, a_2, \ldots} \in X$.

We have that:

Hence:


 * $L \circ R = I$

However:

Therefore, $T \circ S = I$ but $S \circ T \ne I$.