Definition. $L(X, Y)$, products and norms [br96]

    1. Let $L(X, Y)$ be the set of all linear transformations of the vector space $X$ into the vector space $Y$. Instead of $L(X, X)$, we shall simply write $L(X)$.

    2. If $X, Y, Z$ are vector spaces, and if $A \in L(X, Y)$ and $B \in L(Y,Z)$, we define their product $BA$ to be the composition of $A$ and $B$: $$(BA)\mathbf{x} = B(A\mathbf{x}) \qquad (\mathbf{x} \in X).$$ Then $BA \in L(X, Z)$.

    3. For $A \in L(\mathbb{R}^{n}, \mathbb{R}^{m})$, define the norm $\lVert A \rVert$ of $A$ to be the sup of all numbers $\lvert A\mathbf{x} \rvert$, where $\mathbf{x}$ ranges over all vectors in $\mathbb{R}^{n}$ with $\lvert \mathbf{x} \rvert \leq 1$.

      Observe that the inequality $$\lvert A\mathbf{x} \rvert \leq \lVert A \rVert \lvert \mathbf{x} \rvert $$ holds for all $\mathbf{x} \in \mathbb{R}^{n}$. Also, if $\lambda$ is such that $\lvert A\mathbf{x} \rvert \leq \lambda \lvert \mathbf{x} \rvert$ for all $\mathbf{x} \in \mathbb{R}^{n}$, then $\lVert A \rVert \leq\lambda$.