Proof of L²(μ) Is a Real Hilbert Space

Throughout, $(L^{2}(\mu),\|\cdot\|_{2})$ is a real normed space by The Riesz–Fischer Theorem (i), every element of $L^{2}(\mu)$ is $[f]$ for some $f\in\mathcal{L}^{2}(\mu)$ (Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (iv), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (v)), and for $f,g\in\mathcal{L}^{2}(\mu)$ the product $fg$ lies in $\mathcal{L}^{1}(\mu)$ (Hölder's Inequality (ii)). Let $f,g,h\in\mathcal{L}^{2}(\mu)$ and $\lambda\in\mathbb{R}$.

The square of the norm. For $x\in X$, $f(x)f(x)=|f(x)|^{2}$, the real power with exponent $2$ agreeing with the natural power (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (xii), Properties of Real Powers (i)). So $ff$ is the function $|f|^{2}$, and by The Lebesgue Space Lᵖ(μ) (iv), Elementary Properties of the p-Seminorm (ii) and The Lebesgue Space Lᵖ(μ) (iii),

$$\langle[f],[f]\rangle=\int|f|^{2}\,d\mu=\|f\|_{2}^{2}=\|[f]\|_{2}^{2}.$$

An inner product space. By The Lebesgue Space Lᵖ(μ) (ii), The Lebesgue Space Lᵖ(μ) (iv), the pointwise identities $fg=gf$, $(f+g)h=fh+gh$ and $(\lambda f)g=\lambda(fg)$, and Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii),

$$\langle[f],[g]\rangle=\int fg\,d\mu=\int gf\,d\mu=\langle[g],[f]\rangle,$$ $$\langle[f]+[g],[h]\rangle=\int(f+g)h\,d\mu=\int(fh+gh)\,d\mu=\langle[f],[h]\rangle+\langle[g],[h]\rangle,$$ $$\langle\lambda[f],[g]\rangle=\int(\lambda f)g\,d\mu=\int\lambda(fg)\,d\mu=\lambda\,\langle[f],[g]\rangle .$$

By the previous step, $\langle[f],[f]\rangle=\|[f]\|_{2}^{2}\ge0$; and if $\langle[f],[f]\rangle=0$, then $\|[f]\|_{2}=0$, so $[f]$ is the zero vector by Norms and Real Normed Spaces (ii). These are the conditions of Real Inner Product Spaces (i), Real Inner Product Spaces (ii), Real Inner Product Spaces (iii) and Real Inner Product Spaces (iv), so $L^{2}(\mu)$ is a real inner product space (Real Inner Product Spaces (vi)). This proves (i).

The induced norm. The number $\|[f]\|_{2}$ is nonnegative (Norms and Real Normed Spaces (i)) and its square is $\langle[f],[f]\rangle$, so it is the square root of $\langle[f],[f]\rangle$ by the uniqueness in The k-th Root and the Square Root of a Nonnegative Real Number (i) and The k-th Root and the Square Root of a Nonnegative Real Number (ii). By Real Inner Product Spaces (vii) this is (ii).

A Hilbert space. Assume the axiom of dependent choice. By (ii), $L^{2}(\mu)$ with its induced norm, as regarded in Real Hilbert Spaces (i), is $(L^{2}(\mu),\|\cdot\|_{2})$, which is a real Banach space by The Riesz–Fischer Theorem (iv). Hence $L^{2}(\mu)$ is a real Hilbert space. This proves (iii).