关于Stein实分析中线性泛函与内积相关定理里$||l|| = ||g||$的等式推导疑问

Hey Jason, let's work through this equality step by step—you already got the Cauchy-Schwarz part right, so we just need to fill in the last pieces to connect to $||l|| = ||g||$!

First, let's recap the key definitions to make sure we're on the same page:

• For a bounded linear functional $l: H \to \mathbb{C}$ (since we're dealing with inner products, $H$ is a Hilbert space here), its norm is defined as $||l|| = \sup\left{ |l(f)| \mid ||f|| \leq 1, f \in H \right}$. This is a special case of the bounded operator norm you listed, tailored to functionals that map to the scalar field.
• You already applied Cauchy-Schwarz to get $|l(f)| = |\langle f, g \rangle| \leq ||f|| \cdot ||g||$ for all $f \in H$.

Let's split the proof of $||l|| = ||g||$ into two directions:

1. Prove $||l|| \leq ||g||$ #

Take any non-zero $f \in H$. Divide both sides of the Cauchy-Schwarz inequality by $||f||$:
$$\frac{|l(f)|}{||f||} \leq ||g||$$
The left-hand side is exactly the ratio we take the supremum of to get $||l||$. Since this holds for all non-zero $f$, taking the supremum over all such $f$ gives:
$$||l|| = \sup_{f \neq 0} \frac{|l(f)|}{||f||} \leq ||g||$$

2. Prove $||l|| \geq ||g||$ #

Now let's get the reverse inequality. If $g = 0$, then $l$ is the zero functional (since $l(f) = \langle f, 0 \rangle = 0$ for all $f$), so $||l|| = 0 = ||g||$ and we're done.

If $g \neq 0$, let's plug $f = g$ into the functional. We get:
$$l(g) = \langle g, g \rangle = ||g||^2$$
By the definition of the functional norm, we know $|l(g)| \leq ||l|| \cdot ||g||$ (this is just applying the boundedness condition to $f = g$). Substitute $|l(g)| = ||g||^2$ into this inequality:
$$||g||^2 \leq ||l|| \cdot ||g||$$
Since $g \neq 0$, we can divide both sides by $||g||$, which gives:
$$||g|| \leq ||l||$$

Putting it all together #

We've shown both $||l|| \leq ||g||$ and $||g|| \leq ||l||$, so by equality of real numbers, $||l|| = ||g||$.

Just to tie this back to Lemma 5.1 you mentioned: for functionals (which are operators to $\mathbb{C}$), that lemma simplifies to $||l|| = \sup{ |l(f)| \mid ||f|| \leq 1 }$, which aligns perfectly with the definition we used here—so this derivation fits right in with Stein's framework.

备注：内容来源于stack exchange，提问作者jason 1