The space $L^2$ is distinguished among the $L^p$ spaces because its norm comes from an inner product. Orthogonality, projection, Fourier coefficients, and least-squares approximation are consequences of this geometry.

Hilbert-space structure

A complex inner-product space $H$ has a map $\langle\cdot,\cdot\rangle:H\times H\to\mathbb C$ satisfying linearity in one argument, conjugate symmetry, and positive definiteness. The induced norm is

\[\|u\|=\sqrt{\langle u,u\rangle}.\]

A Hilbert space is an inner-product space complete in this norm.

For a measure space $(X,\mathcal M,\mu)$,

\[L^2(\mu) =\left\{f:\int_X|f|^2\,d\mu<\infty\right\}/\!\sim\]

is a Hilbert space with

\[\langle f,g\rangle =\int_X f(x)\overline{g(x)}\,d\mu(x).\]

The Cauchy–Schwarz inequality

\[|\langle f,g\rangle| \le\|f\|_2\|g\|_2\]

ensures that the inner product is finite and continuous.

Projection theorem

Let $M\subset H$ be a nonempty closed convex set. For each $u\in H$, there is a unique $m^\ast\in M$ minimizing the distance:

\[\|u-m^\ast\| =\inf_{m\in M}\|u-m\|.\]

If $M$ is a closed linear subspace, the minimizer is characterized by

\[u-m^\ast\perp M.\]

For a finite-dimensional subspace $M=\operatorname{span}{\phi_1,\ldots,\phi_n}$, write

\[m^\ast=\sum_{j=1}^{n}c_j\phi_j.\]

The orthogonality conditions yield the normal equations

\[\sum_{j=1}^{n} \langle\phi_j,\phi_i\rangle c_j =\langle u,\phi_i\rangle, \qquad i=1,\ldots,n.\]

The matrix $G_{ij}=\langle\phi_j,\phi_i\rangle$ is the Gram matrix. An orthonormal basis makes $G=I$; an ill-conditioned basis makes the numerical problem sensitive even though the projection is mathematically unique.

Orthonormal systems

A family $(e_n)$ is orthonormal when

\[\langle e_m,e_n\rangle=\delta_{mn}.\]

For every $u\in H$, Bessel’s inequality gives

\[\sum_n|\langle u,e_n\rangle|^2\le\|u\|^2.\]

If the system is complete, Parseval’s identity holds:

\[\|u\|^2 =\sum_n|\langle u,e_n\rangle|^2,\]

and

\[u=\sum_n\langle u,e_n\rangle e_n\]

with convergence in the Hilbert-space norm.

The convergence mode must be stated. $L^2$ convergence means

\[\left\|u-\sum_{n=1}^{N} \langle u,e_n\rangle e_n\right\|_2\to0.\]

It does not by itself imply pointwise or uniform convergence.

Fourier series in L²

On $(-\pi,\pi)$, the normalized exponentials

\[e_n(x)=\frac{1}{\sqrt{2\pi}}e^{inx}, \qquad n\in\mathbb Z,\]

form an orthonormal basis of $L^2(-\pi,\pi)$. The Fourier coefficient is

\[\widehat f(n) =\langle f,e_n\rangle =\frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi}f(x)e^{-inx}\,dx.\]

The partial sums converge to $f$ in $L^2$, and Parseval gives conservation of squared norm:

\[\|f\|_2^2=\sum_{n\in\mathbb Z}|\widehat f(n)|^2.\]

Stronger convergence requires stronger hypotheses or different summation procedures. A spectral approximation should therefore report whether its guarantee is in mean square, pointwise, or uniform norm.

Sturm–Liouville structure

A regular Sturm–Liouville problem has the form

\[-\frac{d}{dx}\!\left(p(x)y'(x)\right) +q(x)y(x) =\lambda w(x)y(x)\]

on $[a,b]$, together with boundary conditions that make the operator self-adjoint. Here $p>0$ and $w>0$ under the standard regular assumptions.

If $y_m,y_n$ correspond to distinct eigenvalues, integration by parts and the boundary conditions give

\[(\lambda_m-\lambda_n) \int_a^b y_m(x)\overline{y_n(x)}w(x)\,dx=0.\]

Hence the eigenfunctions are orthogonal in the weighted space $L^2((a,b),w(x)\,dx)$, or equivalently in the Stieltjes space with $d\alpha=w\,dx$. Completeness and spectral behavior require the precise operator domain and endpoint hypotheses; formal orthogonality alone is not a spectral theorem.

Riesz representation

Every bounded linear functional $\Lambda:H\to\mathbb C$ has a unique representing vector $g\in H$ such that

\[\Lambda(f)=\langle f,g\rangle, \qquad \|\Lambda\|=\|g\|.\]

This identifies dual quantities with vectors in the same Hilbert space. In weak formulations, it converts a continuous linear load functional into an inner-product representation whenever the selected Hilbert structure permits it.

CAD and computational geometry

Hilbert-space projection is the mathematical core of many engineering algorithms:

  • fitting a curve or surface in a finite basis;
  • minimizing mean-square geometric deviation;
  • projecting residuals onto trial spaces;
  • spectral decomposition of parameterized fields;
  • reduced-order approximation.

The mathematics also exposes the limits. A least-squares optimum minimizes the declared $L^2$ error, not necessarily the maximum deviation. Parameter-space $L^2$ fitting is not invariant under reparameterization unless the measure is transformed appropriately. For a surface, using physical area measure rather than uniform parameter measure prevents the basis fit from being biased by parameter stretching.

Professional use of $L^2$ therefore specifies the measure, the subspace, the convergence norm, and the conditioning of the chosen basis.