Bounded linear operator

Let $$V_1$$ and $$V_2$$ be normed spaces and let $$T$$ be a linear operator $$T:V_1\to V_2$$. $$T$$ is bounded (see Def 2.4.1 ) if there exists a constant $$K\ge 0$$ such that $$||T\mathbf{v}||_{V_2} \le K||\mathbf{v}||_{V_1}$$ $$\forall \mathbf{v} \in V_1$$.

The smallest possible value of $$K$$ that can be used above is called the norm of the operator $$T$$, and is denoted by $$||T||$$.

Linearity in infinite sums
Note that linearity is in general defined for finite sums. However, for a bounded linear operator it can apply subject to the following convergence criteria (Exercise 2.14 ).

Assume that $$\{\mathbf v_k\}_{k=1}^{\infty}$$ is a sequence of elements in $$V_1$$ and that $$\sum_{k=1}^{\infty} c_k \mathbf v_k$$ is convergent for some scalar sequence $$\{c_k\}_{k=1}^\infty$$. Then,

$$T\sum_{k=1}^\infty c_k \mathbf v_k = \sum_{k=1}^\infty c_k T_k \mathbf v_k$$.