sphinx-proof: advanced proofs, theorems & more…

Easily render proofs, theorems, axioms, lemmas, definitions and much more with sphinx-proof. Read more in the documentation of sphinx-proof.

Proof. We’ll omit the full proof.

But we will prove sufficiency of the asserted conditions.

To this end, let \(y \in \mathbb R^n\) and let \(S\) be a linear subspace of \(\mathbb R^n\).

Let \(\hat y\) be a vector in \(\mathbb R^n\) such that \(\hat y \in S\) and \(y - \hat y \perp S\).

Let \(z\) be any other point in \(S\) and use the fact that \(S\) is a linear subspace to deduce

\[\| y - z \|^2 = \| (y - \hat y) + (\hat y - z) \|^2 = \| y - \hat y \|^2 + \| \hat y - z \|^2\]

Hence \(\| y - z \| \geq \| y - \hat y \|\), which completes the proof.