Space of Bounded Linear Transformations forms Vector Space

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.

Let $\map B {X, Y}$ be the space of bounded linear transformations between $X$ and $Y$.

Let $+_B$ and $\circ_B$ be pointwise addition and pointwise scalar multiplication on $Y^X$.

Then $\struct {\map B {X, Y}, +_B, \circ_B}_\GF$ is a vector space.

Proof
Let $\map L {X, Y}$ be the space of linear transformations between $X$ and $Y$.

From Space of Linear Transformations forms Vector Space, $\map L {X, Y}$ is a vector space over $\GF$ with pointwise addition and pointwise scalar multiplication.

It therefore suffices to show that $\map B {X, Y}$ is a vector subspace of $\map L {X, Y}$.

From One-Step Vector Subspace Test, it suffices to show that $\map B {X, Y} \ne \O$ and:


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

for each $\lambda \in \GF$ and $T, S \in \map B {X, Y}$.

First, we have:


 * $\norm {\mathbf 0_{\map L {X, Y} } x}_Y = 0 \le \norm x_X$

So $\mathbf 0_{\map L {X, Y} } \in \map B {X, Y}$.

Now let $T, S \in \map B {X, Y}$.

Then there exists $M, M' > 0$ such that:


 * $\norm {T x}_Y \le M \norm x_X$ for each $x \in X$

and:


 * $\norm {S x}_Y \le M' \norm x_X$ for each $x \in X$.

Then for each $\lambda \in \GF$ and $x \in X$ we have:

So $T + \lambda S \in \map B {X, Y}$.