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 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 $$||T||$$.