Bounded Operator Norm: Special Element

Question

Given a Banach spaces $X$ and $Y$.

Consider a bounded operator: $$T:X\to Y:\quad\|T\|<\infty$$

Then theres an element: $$\|Tx\|=\|T\|\cdot\|x\|\quad(x\neq0)$$ Does it always exist?

Answer 1

How about this: define $T : \ell^1 \to \ell^1$ by

$$T(a_1, a_2, a_3, \cdots, a_n, \cdots) = (0, (1-1/2)a_2, (1-1/3)a_3, \cdots, (1-1/n)a_n, \cdots)$$

Then $$||T|| = \sup_{||x|| = 1} ||Tx|| = \sup_n (1-1/n) = 1$$

However, for all non-zero $x \in \ell^1$, $||Tx|| < ||x|| = ||T|| \cdot ||x||$.