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This point has been giving me a lot of trouble. I am skipping around in learning functional analysis, and I've directly gone to the study of bounded operators without studying topology and basic Banach Space theory, in hopes to pick it up along the way. I'm doing okay with this....

Anyways, I am trying to show that given two Banach Spaces $X$ and $Y$, the operator $T$ in $L(X,Y)$ and its adjoint $T^{\ast}$ in $L(Y^{\ast},X^{\ast})$, then the map $T \mapsto T^\ast$ is an isometric isomorphism of $L(X,Y)$ into $L(Y^\ast,X^\ast)$. The proof is very straightforward, with one important detail, which is

$$\Vert Tx\Vert_Y = \sup_{\Vert l \Vert \leq 1} |l(Tx)| \qquad \text{for } l \in Y^{\ast}$$

the authors justify this as follows: "this equality uses a corollary of the Hahn-Banach Theorem".

I just don't see this! How does this follow?

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up vote 3 down vote accepted

One important consequence of the Hahn-Banach theorem is the following:

If $x \in X$ is a non-zero vector then there exists a functional $\phi:X \to \mathbb{R}$ of norm $1$ such that $\phi(x) = \|x\|$. (This is of course trivial for $x = 0$)

To see this, take the linear subspace $U$ generated by $x$, define $\varphi: U \to \mathbb{R}$ by $\varphi(x) = \|x\|$, $\varphi(\lambda x) = \lambda\|x\|$. Note that $\varphi:U \to \mathbb{R}$ is a functional satisfying $\|\varphi\| = 1$ and extend it linearly and norm-preservingly to a functional $\phi: X \to \mathbb{R}$.

One application of this observation is that the canonical inclusion $X \to X^{\ast\ast}$ is an isometry.

I interpret your formula as $$\|Tx\|_Y = \sup_{\|\ell\| \leq 1} |\ell(Tx)|$$ which is then an immediate consequence of the highlighted comment above.

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@ Theo- thanks again, you are very helpful in helping me understand this subject! – r.g. Apr 22 '11 at 5:41
@Rohan: No problem at all, it's my pleasure. It takes some time to get used to the standard gloss in functional analysis (it took me a while to be sure). Don't worry too much about not knowing a lot about topology. As others have already suggested, take the time to read the first few sections of Pedersen's Analysis NOW, where you can pick up all you'll ever need for learning functional analysis. It is not a leisurely read, but definitely worth the effort. – t.b. Apr 22 '11 at 5:48
@Rohan: One more thing: Have you noticed that you can accept answers by clicking on the gray checkmark sign on the left? This would prevent the questions to be bumped to the front page every once in a while in the future. – t.b. Apr 22 '11 at 5:50
Questions aren't bumped if they have answers with positive vote count. In cases of good answers that don't have positive vote count, there can be a positive side effect to not accepting, because the bumping allows more users to see the answer and hopefully upvote it. Of course, that's not relevant to this good answer which does have a positive vote count. – Jonas Meyer Apr 22 '11 at 18:28
@Jonas: Thanks! I knew there was a certain threshold to bumping, but I wasn't sure anymore (I thought it was two votes on the answers in total). Moreover, it was the third question he didn't vote on nor accept, so I thought I might point out the site's workings. – t.b. Apr 22 '11 at 18:38

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