Continuous Linear Transformations form Subspace of Linear Transformations

Theorem
The space of continuous linear transformations is a subspace of the space of linear transformations.

Proof
Let $\struct {X, \norm \cdot }$ and $\struct {Y, \norm \cdot }$ be normed vector spaces.

Closure under vector addition
Let $S, T \in \map {CL} {X, Y}$.

By continuity of linear transformations in normed vector space:


 * $\exists M_S \in \R : M_S > 0 : \forall x \in X : \norm {S x} \le M_S \norm x$


 * $\exists M_T \in \R : M_T > 0 : \forall x \in X : \norm {T x} \le M_T \norm x$

Furthremore:

By continuity of linear transformations in normed vector space:


 * $S + T \in \map {CL} {X, Y}$

Closure under scalar multiplication
Let $\alpha \in \R$.

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

By continuity of linear transformations in normed vector space:


 * $\exists M \in \R_{> 0} : \forall x \in X : \norm {T x} \le M \norm x$

Hence:

By continuity of linear transformations in normed vector space:


 * $\alpha T \in \map {CL} {X, Y}$

Existence of identity element under vector addition
Let $\mathbf 0 : X \to Y$ be the zero mapping.

Then:

Hence:


 * $\mathbf 0 \in \map {CL} {X, Y}$