Currently, I'm studying the relationship btw linear functional and inner product, and this is the associated theorem in Stein's Real Analysis. (theorem 5.3, chapter 4)
I understand all of the proofs, except for the red line $||l|| = ||g||$. I'm not sure how the equality holds. I tried to apply Cauchy-Schwartz inequality $|(l(f)| = |(f,g)| \le ||f||||g||$, but I failed to extend to the conclusion.
Any idea about this question would be grateful. Thank you.
For reference, let me write the definition of some concepts in Stein's book.
A linear operator $T : H_1 \to H_2$ is bounded if there is $M > 0$ such that $||T(f)||_{H_2} \le M||f||_{H_1}$ $\forall f \in H_1$
The norm of $T$ is defined by $||T|| = inf\{M\; |\; ||T(f)||_{H_2} \le M||f||_{H_1}\}$
Lemma 5.1 $||T|| = sup\{|(T(f), g) \;|\;||f|| \le 1,\; ||g|| \le 1,\; f \in H_1, g \in H_2 \}$
Suppose $g \neq 0$. $|\ell (f)| \leq \|f\|\|g\|$ so $\|\ell\|\leq \|g\|$. Also, $\ell (\frac g {\|g\|}) =\langle \frac g {\|g\|}, g \rangle =\|g\|$ and $\|\frac g {\|g\|}\|=1$ so $\|\ell \|\geq \|g\|$. The case $g=0$ is trivial.