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A convex set that is also a closed subset in the specified topology. In a Hilbert space with its norm topology, every nonempty closed convex set has a unique nearest point to every ambient vector, by the Hilbert projection theorem.

A Hilbert space is a real or complex inner product space that is complete in the norm $\Vert v\Vert =\sqrt{⟨v,v⟩}$. Completeness means that every Cauchy sequence has a limit in the space: if its terms become arbitrarily close in this norm, they converge to an allowed vector. The inner product supplies lengths and orthogonality; completeness ensures that limiting approximation procedures do not leave the space.

Finite-dimensional Euclidean spaces are examples, but the same geometry also works for sequences and functions. In the l2 sequence space,In the L2 space on $[-1,1]$ with Lebesgue measure,These formulas use the convention linear in the first argument. Elements of the L2 space are equivalence classes of square-integrable functions agreeing almost everywhere, so zero distance really identifies the same element. A vector can therefore be a whole function, and its squared length can measure an integrated squared error.

For example, approximating $f(x)=x^2$ by $a+bx$ is a nearest-point problem in this Hilbert space. The allowed approximations form the linear span of $1,x$, and the orthogonal projection is the constant $1/3$. Its residual $x^2-1/3$ is orthogonal to both allowed directions because its integral and its inner product with $x$ vanish. The geometry therefore identifies the best approximation without searching through every pair of coefficients.

This is useful whenever a complicated vector must be represented by a tractable collection of directions. Fourier series approximate functions with orthogonal oscillations; the Parseval identity relates the sum of squared coefficients to the squared norm. Least-squares solutions approximate attainable data using a linear operator. The Hilbert projection theorem explains why a nearest element exists and is unique for a nonempty closed convex set. Distinct application settings share the same geometry of useful components and orthogonal residuals. Casey Rodriguez's MIT lectures develop these connections between Hilbert spaces, Fourier series, and nearest-point problems.^[ref]

A historical mathematical setting is David Hilbert's Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1912). Its preface states that it reprints six communications published during 1904–1910 on linear integral equations. This provides a concrete source for the integral-equation setting associated with the name, while the abstract definition above incorporates later developments.^[ref]

The parallelogram law for a quadratic form $Q$ isConversely, under the usual divisibility assumptions, a function satisfying this identity and the appropriate homogeneity comes from a symmetric bilinear form by polarization.

For an inner product space, taking $Q(v)=\Vert v{\Vert }^2$ gives the displayed squared-norm identity by expanding and cancelling the mixed inner products. It is the geometric ingredient that makes minimizing sequences converge in the Hilbert projection theorem.