Unique Point of Minimal Distance to Closed Linear Subspace of Hilbert Space iff Orthogonal

Theorem

Let $H$ be a Hilbert space, and let $h \in H$.

Let $K \subseteq H$ be a closed linear subspace of $H$.

Then the unique point $k_0 \in K$ from Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space such that:

$\norm {h - k_0} = \map d {h, K}$

where $d$ denotes distance to a set, is characterised by:

$\norm {h - k_0} = \map d {h, K} \iff \paren {h - k_0} \perp K$

where $\perp$ signifies orthogonality.

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Theorem $2.6$