---- > [!theorem] Theorem. ([[Hilbert projection theorem]]) > Suppose $X$ is a [[Hilbert space]], $f \in X$, and $A$ is a nonempty [[closed set|closed]] [[convex set|convex]] subset. Then there exists a unique [[distance from point to set|projection]] $\hat{f} \in A$, i.e. a unique $\hat{f} \in A$ such that $\|f-\hat{f}\|=\text{dist}(x, A).$ ^theorem > [!specialization] > - For [[dimension|finite dimensions]]: [[minimum distance to a subspace is the orthogonal projection onto it]] ^specialization > [!note] Remark. > Both existence and uniqueness of the projection can fail if we only assume $X$ is a [[Banach space]]. ^note > [!proof]- Proof. ([[Hilbert projection theorem]]) > Omitted. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```