V a IPS of finite dimension, $\phi : V \to F$ is a linear functional. Let $B = \{v_1...v_n\}$ an orthonormal basis for V.

Question

1. Prove Riesz unique representation theorom:

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If $v \in V$ is a vector in $V \implies \exists ! u \in V$ that stasfies $\phi (v) = \;<v,u>$

$\mathbf {Hint}$: express $u$ as a linear combination of $B$ and fix the coefficients.

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2. Prove if $\phi \neq 0 \implies \forall v \in V$ we have:

$d(v,ker \phi) = {|\phi (v)|\over {\sqrt{\phi (u)}}}$

$\mathbf {Whilst}\;$ $d(v, W)$ is the distance between $V$ and $W \subseteq V$, or in other words $||v-p_w(v)||$.

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So I found proof of the first part:

https://i.stack.imgur.com/Iz4Y7.png

https://i.stack.imgur.com/HCIIL.png

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But I would love to understand the third paragraph because I don't understand where the vector $v - {\phi (v)u\over ||u^2||}$ came from and why $\phi (v-{\phi (v)u\over ||u^2||}) = 0$.

Regarding the second part I'm would love ideas, because I'm very much lost on it.