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

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, write $\mathcal{L}^{2}$ for the set of $2$-integrable functions on $(X,\mathcal{F},\mu)$, and let $L^{2}=L^{2}(X,\mathcal{F},\mu)$ be the Lebesgue space of classes of such functions, a real vector space by The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §normed. Then the following hold.

1. (The inner product)¶ For $f,g\in\mathcal{L}^{2}$ the pointwise product $fg$ is integrable, and the real number $\int_{X}fg\,d\mu$ depends only on the classes $[f]$ and $[g]$. The map

is an inner product on $L^{2}$, so that $L^{2}$ is a real inner product space. Its norm is the norm of The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §normed: for every $f\in\mathcal{L}^{2}$ the norm of $[f]$ equals $\lVert[f]\rVert_{2}$, and the distance of the inner product space coincides with the distance $d_{2}$ of that claim.

2. (A real Hilbert space)¶ $L^{2}$, with the inner product of claim 1, is a real Hilbert space.