Norm of a linear map: Alternative definition?

Question

Let $T: V \to W$ be a bounded linear map between normed spaces. I'm wondering if it is true that

$$\Vert T \Vert = \sup\{\Vert Tv\Vert : v \in V, \Vert v \Vert < 1\}$$

Note the strict inequality. In the usual definition this is $\leq $.

I could not prove it nor provide a counterexample.

Answer 1

Let's first look at the usual definition - if the norm of $T$ is $M$, then $||Tv||\leq M||v||$ for all unit vectors, and there is a sequence of unit vectors $v_n$ s.t $||Tv_n||\rightarrow M$.

Now, we can look at the sequence $\frac{n\cdot v_n}{n+1}$. These are vectors of norm less than $1$. What does the norm of this sequence converge to?