$||\phi||=1$ and $|\phi (x)|=||x||$

Question

a.Let $E$ be a non-zero Banach space and show that for every $x\in E$ there is $\phi \in E^*$ such that $||\phi ||=1$ and $|\phi (x)|=||x||$

b. Let E and F be Banach spaces,let $\pi: E\to F $ be a bounded linear map and let $ \pi^*:F^* \to E^* $ be the induced map on dual spaces. Show that $||\pi^*||=||\pi||$.

No idea how should i start it. I don't know why it seems Banach space is very difficult for me. I am sure i have to do lots of problems from Banach space before my qualifying exam. Please help me.

Answer 1

b) $$ \|\pi \| = \sup_{x \neq 0} \dfrac{\| \pi(x) \|}{\| x \|} = \sup_{x \neq 0}\sup_{\phi \neq 0} \dfrac{ | \phi( \pi(x))| }{\|x\| \|\phi \|} = $$ $$ \sup_{x \neq 0}\sup_{\phi \neq 0} \dfrac{ | \phi( \pi(x))| }{\|x\| \|\phi \|} = \sup_{x \neq 0}\sup_{\phi \neq 0} \dfrac{ | (\pi^* \phi)(x)| }{\|x\| \|\phi \|} = \sup_{\phi \neq 0} \dfrac{ \| \pi^* \phi \|}{\| \phi \|} =\| \pi^* \| $$

the second = is an application of a corollary of HB; The following = is the definition of adjoint. The second to last equality is the "most tricky", try to work it out :) (Hint: think $x$ as a functional and apply the definition of norm)

Answer 2

To do (a), try applying the Hahn-Banach extension theorem.