Proof of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space

Each result cited is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement above.

Claim 1. Since $\tfrac12+\tfrac12=1$ and $1<2$, the number $2$ is its own conjugate exponent, by the uniqueness in Conjugate Exponents and Young's Inequality (i). Let $f,g\in\mathcal{L}^{2}$. By Hölder's Inequality (i) the pointwise product $fg$ lies in $\mathcal{L}^{1}$, that is, $fg$ is measurable with $\int_{X}|fg|\,d\mu<\infty$, and so is integrable by the criterion recorded in Measurable Functions and the Lebesgue Integral: Standing Notation (v).

If $f\sim f'$ and $g\sim g'$, then the set where $fg$ and $f'g'$ differ is contained in the union of the set where $f$ and $f'$ differ and the set where $g$ and $g'$ differ, hence is null by The Lebesgue Integral and Null Sets (i); so $fg=f'g'$ almost everywhere and $\int_{X}fg\,d\mu=\int_{X}f'g'\,d\mu$ by The Lebesgue Integral and Null Sets (iii). Thus the displayed formula depends only on the classes and defines a map assigning a real number to each pair of elements of $L^{2}$.

We check the four conditions of Real Inner Product Spaces (i), using that $L^{2}$ is a real vector space by The Riesz–Fischer Theorem (i) and that its operations are computed on representatives.

(a) Symmetry holds because $fg=gf$ pointwise.

(b) For $f,g,h\in\mathcal{L}^{2}$ the products $fh$ and $gh$ are integrable by the first paragraph, and $(f+g)h=fh+gh$ pointwise, so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives $\int_{X}(f+g)h\,d\mu=\int_{X}fh\,d\mu+\int_{X}gh\,d\mu$, which is additivity in the first argument.

(c) Likewise $(cf)h=c(fh)$ pointwise, so $\int_{X}(cf)h\,d\mu=c\int_{X}fh\,d\mu$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion.

(d) For $f\in\mathcal{L}^{2}$ we have $f(x)f(x)=(|f(x)|)^{2}$ for every $x$, the power with exponent $2$ agreeing with the natural power by Properties of Real Powers (ii). Hence the integrand $ff$ is nonnegative, so $0\le\int_{X}ff\,d\mu$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion applied to the constant map $0_{X}$ and $ff$, and

$$\bigl\langle[f],[f]\bigr\rangle_{L^{2}}=\int_{X}|f|^{2}\,d\mu .$$

If this vanishes then $\lVert f\rVert_{2}=0^{1/2}=0$, so $f=0$ almost everywhere by Elementary Properties of the p-Seminorm (v) and $[f]$ is the zero vector of $L^{2}$, identified in The Riesz–Fischer Theorem (i). This is positive definiteness, and $L^{2}$ is a real inner product space.

The norm of $[f]$ is the unique nonnegative real number whose square is $\langle[f],[f]\rangle_{L^{2}}=\int_{X}|f|^{2}\,d\mu$. The number $\lVert[f]\rVert_{2}=(\int_{X}|f|^{2}\,d\mu)^{1/2}$ is nonnegative and its square is $\int_{X}|f|^{2}\,d\mu$ by Elementary Properties of the p-Seminorm (i); by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root the two agree. Consequently the distance of the inner product space, namely the norm of $[f]-[g]$, equals $\lVert[f]-[g]\rVert_{2}=d_{2}([f],[g])$ as computed in The Riesz–Fischer Theorem (i).

Claim 2. By claim 1 the metric of the real inner product space $L^{2}$ is $d_{2}$. By The Riesz–Fischer Theorem (ii) the space $L^{2}$ is a real Banach space, that is, by Norms and Real Normed Spaces (v), the metric space $(L^{2},d_{2})$ is complete. Hence $L^{2}$ is a real Hilbert space by Real Hilbert Spaces (i).