Vector Space of Bounded Linear Operators

We can also equip this space with a norm, called the operator norm.

Proof

Using the vector subspace test, we simply must prove that $B(X,Y)$ is a non-empty set, closed under addition and scalar multiplication.

It is clear that the zero map is in $B(X,Y)$, since zero is bounded by any non-negative real number. Suppose $T,S\in B(X,Y)$, and $\lambda \in \mathbb{R}$. This means, there exists constants $C_1$ and $C_2$ such that

for all $\mathit{x}\in X$.

Therefore

and