L²(μ) Is a Real Hilbert Space

In the settings of Measurable Functions and the Lebesgue Integral: Standing Notation and Real Normed, Inner Product and Hilbert Spaces: Notation, let $(X,\mathcal{F},\mu)$ be a measure space. Let $L^{2}(\mu)$ be the Lebesgue space with $p=2$, with the addition and scalar multiplication of The Lebesgue Space Lᵖ(μ) (ii), the norm $\|\cdot\|_{2}$ of The Lebesgue Space Lᵖ(μ) (iii) and the inner product $\langle\cdot,\cdot\rangle$ of The Lebesgue Space Lᵖ(μ) (iv).

(i) An inner product space. $L^{2}(\mu)$, with $\langle\cdot,\cdot\rangle$, is a real inner product space.

(ii) The induced norm. The induced norm of $L^{2}(\mu)$ is $\|\cdot\|_{2}$:

$$\sqrt{\langle F,F\rangle}=\|F\|_{2}\qquad(F\in L^{2}(\mu)).$$

(iii) A Hilbert space. Assume the axiom of dependent choice. Then $L^{2}(\mu)$ is a real Hilbert space.