Vector Space of Bounded Linear Operators

Definition

The set of bounded linear operators between normed vector spaces \(X\) and \(Y\) is denoted by \(B(X, Y)\).

Theorem

\(B(X, Y)\) forms a vector space under pointwise addition and multiplication.

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

\[ \|T(\boldsymbol{x})\| \leq C_1 \|\boldsymbol{x}\| \quad \text{and} \quad \|S(\boldsymbol{x})\| \leq C_2 \|\boldsymbol{x}\|\]

for all \(\boldsymbol{x} \in X\).

Therefore

\[\begin{align*} \|T(\boldsymbol{x}) + S(\boldsymbol{x})\| &\leq \|T(\boldsymbol{x})\| + \|S(\boldsymbol{x})\| \\ &\leq C_1 \|\boldsymbol{x}\| + C_2 \|\boldsymbol{x}\| \\ &\leq (C_1 + C_2) \|\boldsymbol{x}\| \\ \end{align*}\]

and

\[\begin{align*} \|\lambda T(\boldsymbol{x})\| &\leq (C_1 \lambda) \|T(\boldsymbol{x})\|. \end{align*}\]