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So I’m pondering the statement:

Show that a continuous linear functional $f$ on a subspace $V$ of a Hilbert space $H$ has a unique norm preserving extension $h$ on $H$

Here is my thought.

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- Application of a general “Weierstrass theorem”
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- Norms of linear maps
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Consider the closure of $V$, $\overline{V}$. Then since $V$ is dense in $\overline{V}$, there exists a unique extension $\overline{f}$ of $f$ to $\overline{V}$.

Moreover since $\overline{V}$ is a closed subspace of $H$ then it is also a Hilbert space, so by the Frechet-Riesz theorem we have $\overline{f}(v)$ = $\langle v,y_{\space\overline{f}} \rangle$ for some unique element $y_{\space\overline{f}} \in \overline{V}$

Take $ h(x) = \langle x,y_{\space\overline{f}} \rangle$.

How do I get uniqueness though?

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- Fourier transform in $L^p$
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- Why don't analysts do category theory?

Let me simplify things by assuming (as you point out, without loss of generality) that $V$ is closed. Then, indeed, $f(x) = \langle x,y \rangle$ on $V$ for a unique $y \in V$, yielding the immediate extension $h(x) := \langle x,y\rangle$ of $f$ to $H$; one can readily check that $$\|f\|_{V^\ast} = \|y\|_V = \|y\|_H = \|h\|_{H^\ast},$$ so that this extension is norm-preserving.

Now, suppose that $k \in H^\ast$ is another extension of $f$ to $H$, so that $k(x) = \langle x,z\rangle$ for a unique $z \in H$. Then for any $x \in V$, $$0 = f(x) – f(x) = h(x) – k(x) = \langle x,y\rangle – \langle x,z\rangle = \langle x,y-z\rangle,$$ so that $y-z \in V^\perp$. Hence, $y$ and $y-z$ are orthogonal, so that

$$\|k\|_{H^\ast} = \|z\|_H = \|y – (y-z)\|_H = \sqrt{\|y\|^2_H + \|y-z\|^2_H}.$$ Thus, $\|k\|_{H^\ast} = \|f\|_{V^\ast}$ if and only if $\sqrt{\|y\|^2_H + \|y-z\|^2_H} = \|y\|_H$, if and only if $\|y-z\| = 0$, if and only if $y =z$, if and only if $h=k$. Hence, $h$ is indeed the unique norm-preserving extension.

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