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We will now revisit “The Fundamental Theorem of Approximation” for the extremely important case where our set $A$ is a subspace. Specifically, suppose that $H$ is a Hilbert space, and let $A$ be a (closed) subspace of $H$ . From before, we have that for any $x\in H$ there is a unique $\widehat{x}\in A$ such that $\widehat{x}$ is the closest point in $A$ to $x$ . When $A$ is also a subspace, we also have:
$\widehat{x}\in A$ is the minimizer of $\u2225x,-,\widehat{x}\u2225$ if any only if $\widehat{x}-x\perp A$ i.e., $\u2329\widehat{x},-,x,,,y\u232a=0$ for all $y\in A$ .
Let $z=\widehat{x}+\delta y$ .
Thus $\u2225x,-,z\u2225\le \u2225x,-,\widehat{x}\u2225$ , contradicting the assumption that $\widehat{x}$ minimizes $\u2225x,-,\widehat{x}\u2225.$
This result suggests a that a possible method for finding the bestapproximation to a signal $x$ from a vector space $V$ is to simply look for a vector $\widehat{x}$ such that $\widehat{x}-x\perp V$ . In the coming lectures we will show how to do this, but it will require a brief review of some concepts from linear algebra.
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