L-squared and Completeness

A Hilbert space needs two things, an inner product and completeness. The inner product supplies the geometry, and completeness supplies the limits, without which an orthogonal series has no vector to converge to. This post produces the two concrete Hilbert spaces the rest of the curriculum runs on, the square-integrable functions $L^2(\mu)$ and the square-summable sequences $\ell^2$ , and proves the one nontrivial fact about them, that they are complete. The proof rests on the convergence theorems of measures and integration [1].

#The space of square-integrable functions

Fix a measure space $(\Omega,\mathcal A,\mu)$ . A measurable function $f$ is square-integrable when $\int\abs{f}^2\,d\mu<\infty$ , and $L^2(\mu)$ is the set of such functions with two identified when they agree almost everywhere, so its elements are equivalence classes modulo null sets. It is a vector space, because $\abs{f+g}^2\le 2\abs{f}^2+2\abs{g}^2$ keeps a sum square-integrable, and over the real scalars it carries the candidate inner product

\ip{f}{g}=\int_\Omega fg\,d\mu,\qquad \norm{f}_2=\Big(\int_\Omega\abs{f}^2\,d\mu\Big)^{1/2}. \tag{1}

Proposition1

The inner product Equation (1) is finite for all $f,g\in L^2(\mu)$ , and it makes $L^2(\mu)$ an inner product space.

Proof

Finiteness is the pointwise bound $\abs{fg}\le\half(\abs{f}^2+\abs{g}^2)$ , which integrates to $\int\abs{fg}\,d\mu\le\half(\norm{f}_2^2+\norm{g}_2^2)<\infty$ , so $\int fg\,d\mu$ is a well-defined finite number. Once it exists, the abstract Cauchy-Schwarz inequality sharpens this to $\int\abs{fg}\,d\mu\le\norm{f}_2\norm{g}_2$ . Bilinearity and symmetry are linearity of the integral. Positive definiteness needs the null-set identification, since $\int f^2\,d\mu=0$ forces $f^2=0$ almost everywhere, that is $f=0$ as an element of $L^2(\mu)$ . The form is therefore an inner product. Over the complex scalars one takes $\ip{f}{g}=\int_\Omega f\bar g\,d\mu$ instead, with $\ip{f}{f}=\int\abs{f}^2\,d\mu$ and conjugate symmetry $\ip{g}{f}=\overline{\ip{f}{g}}$ , and the same argument runs with $\abs{f}^2$ in place of $f^2$ .

The sequence space $\ell^2$ is the special case $\Omega=\N$ with counting measure, the square-summable real sequences $x=(x_n)$ with $\sum_n x_n^2<\infty$ and $\ip{x}{y}=\sum_n x_ny_n$ . Everything proved for $L^2(\mu)$ holds for $\ell^2$ by reading the integral as a sum.

#The Riesz-Fischer theorem

The one property that is not formal is completeness. It is a theorem, and it is exactly where the convergence theorems of integration earn their place.

Theorem2

$L^2(\mu)$ is complete. Every Cauchy sequence in the mean-square norm converges to an element of $L^2(\mu)$ .

Proof

Let $(f_n)$ be Cauchy in $L^2(\mu)$ . Choose a subsequence $f_{n_1},f_{n_2},\dots$ that converges fast, $\norm{f_{n_{k+1}}-f_{n_k}}_2\le 2^{-k}$ , which is possible because the sequence is Cauchy. Set

g=\sum_{k=1}^\infty\abs{f_{n_{k+1}}-f_{n_k}}, \tag{2}

a nondecreasing limit of partial sums. The triangle inequality in $L^2$ gives each partial sum norm at most $\sum_k 2^{-k}=1$ , so by the monotone convergence theorem applied to the squared partial sums, $\norm{g}_2\le 1$ and in particular $g<\infty$ almost everywhere. Where $g$ is finite the telescoping series $f_{n_1}+\sum_k(f_{n_{k+1}}-f_{n_k})$ converges absolutely, so $f_{n_k}$ converges pointwise almost everywhere to a limit $f$ , with $\abs{f}\le\abs{f_{n_1}}+g\in L^2(\mu)$ , so $f\in L^2(\mu)$ . Each difference obeys $\abs{f-f_{n_k}}^2\le(\abs{f}+\abs{f_{n_k}})^2\le(2\abs{f_{n_1}} +2g)^2$ , an integrable dominator independent of $k$ , so the dominated convergence theorem gives $\norm{f-f_{n_k}}_2\to 0$ . The subsequence converges in $L^2$ to $f$ , and a Cauchy sequence with a convergent subsequence converges to the same limit, so $f_n\to f$ in $L^2(\mu)$ .

On a measure space carrying two disjoint sets $A,B$ with $0<\mu(A)=\mu(B)<\infty$ , $L^2$ is the only $L^p$ that is a Hilbert space, since by the Jordan-von Neumann characterisation the $L^p$ norm obeys the parallelogram law only at $p=2$ . Taking $f=\mu(A)^{-1/p}\mathbf 1_A$ and $g=\mu(B)^{-1/p}\mathbf 1_B$ gives $\norm{f}_p=\norm{g}_p=1$ and, by disjoint supports, $\norm{f\pm g}_p^p=2$ , so $\norm{f+g}_p^2+\norm{f-g}_p^2=2\cdot 2^{2/p}$ against $2\norm{f}_p^2+2\norm{g}_p^2=4$ , equal iff $p=2$ . On a degenerate space (a single atom) every $L^p\cong\R$ is trivially Hilbert. The Riesz-Fischer theorem makes that inner product space complete, and the combination is the definition of a Hilbert space met in the previous post.

Corollary3

$L^2(\mu)$ and $\ell^2$ are Hilbert spaces.

#Separability

For the expansion theory to come, a Hilbert space should have a countable dense subset, a property called separability, since then it admits a countable orthonormal basis. The space $\ell^2$ is separable, with the finitely supported rational sequences dense in it, and the standard unit vectors $e_n$ , with a one in position $n$ , form an orthonormal basis. The space $L^2(\mu)$ is separable for reasonable measures, for instance Lebesgue measure on a bounded interval $[a,b]$ . There the polynomials with rational coefficients are dense, since $C[a,b]$ is dense in $L^2[a,b]$ , polynomials are sup-norm dense in $C[a,b]$ by Stone-Weierstrass, and $\norm{\cdot}_2\le(b-a)^{1/2}\norm{\cdot}_\infty$ on $[a,b]$ transfers the density. On an unbounded interval no nonzero polynomial lies in $L^2$ , so a different countable family is needed there, for instance step functions on rational subintervals with rational values. Separability is not automatic, but the spaces that arise from processes and operators here all have it, so a countable orthonormal basis is always available. This is the structural fact the Karhunen-Loeve expansion and every Fourier argument exploit.

With the inner product from the previous post and the completeness proved here, $L^2$ is the Hilbert space the analysis of the rest of the curriculum takes place in, the space in which orthogonal expansions converge, conditional expectation projects, and the stochastic integral is defined.

[1]

W. Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991.

Explore connections

see in the atlas

referenced by (6)

cite

@misc{l2-and-completeness,
  author = {Zac Kienzle},
  title  = {L-squared and Completeness},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/l2-and-completeness}
}

#The space of square-integrable functions

\ip{f}{g}=\int_\Omega fg\,d\mu,\qquad \norm{f}_2=\Big(\int_\Omega\abs{f}^2\,d\mu\Big)^{1/2}. \tag{1}

Proposition1

The inner product Equation (1) is finite for all $f,g\in L^2(\mu)$ , and it makes $L^2(\mu)$ an inner product space.

Proof

#The Riesz-Fischer theorem

The one property that is not formal is completeness. It is a theorem, and it is exactly where the convergence theorems of integration earn their place.

Theorem2

$L^2(\mu)$ is complete. Every Cauchy sequence in the mean-square norm converges to an element of $L^2(\mu)$ .

Proof

g=\sum_{k=1}^\infty\abs{f_{n_{k+1}}-f_{n_k}}, \tag{2}

Corollary3

$L^2(\mu)$ and $\ell^2$ are Hilbert spaces.

#Separability

[1]

W. Rudin, Functional Analysis, 2nd ed. McGraw-Hill, 1991.

Explore connections

see in the atlas

referenced by (6)

cite

@misc{l2-and-completeness,
  author = {Zac Kienzle},
  title  = {L-squared and Completeness},
  year   = {2026},
  month  = {05},
  url    = {https://zackienzle.com/blog/l2-and-completeness}
}