= Hilbert space
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A Hilbert space is a real or complex <inner product space> that is <complete> in the <norm> $\|v\|=\sqrt{\langle v,v\rangle}$. 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>,
$$
\langle u,v\rangle=\sum_{n=1}^{\infty}u_n\overline{v_n},\qquad
\|u\|^2=\sum_{n=1}^{\infty}|u_n|^2.
$$
In the <L2 space> on $[-1,1]$ with Lebesgue measure,
$$
\langle f,g\rangle=\int_{-1}^{1}f(x)\overline{g(x)}\,dx,\qquad
\|f\|^2=\int_{-1}^{1}|f(x)|^2\,dx.
$$
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.https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/[Rodriguez, MIT 18.102, lectures 14–17, 2021]{ref}
A historical mathematical setting is David Hilbert's \i[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.https://archive.org/details/grundzugeallg00hilbrich[Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, 1912, preface]{ref}
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