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

Jump to navigation
Jump to search

## 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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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