# Inner Product/Examples/Lebesgue 2-Space

## Example of Inner Product

Let $\tuple {X, \Sigma, \mu}$ be a measure space.

Let $\map {L^2} \mu$ be the Lebesgue $2$-space of $\mu$.

Let $\innerprod \cdot \cdot: \map {L^2} \mu \times \map {L^2} \mu \to \C$ be the mapping defined by:

$\ds \innerprod f g = \int f \, \overline g \rd \mu$

Then $\innerprod \cdot \cdot$ is an inner product on $\map {L^2} \mu$.

## Proof

First of all, by Hölder's Inequality for Integrals with $p = q = 2$, it follows that:

$\ds \int f \, \overline g \rd \mu$

is defined.

Now checking the axioms for an inner product in turn:

### $(1)$ Conjugate Symmetry

 $\ds \innerprod f g$ $=$ $\ds \int f \, \overline g \rd \mu$ $\ds$ $=$ $\ds \int \overline {\overline f \, g} \rd \mu$ Complex Conjugation is Involution $\ds$ $=$ $\ds \overline {\int g \, \overline f \rd \mu}$ $\ds$ $=$ $\ds \overline {\innerprod g f}$

$\Box$

### $(2)$ Sesquilinearity

 $\ds \innerprod {\lambda f + g} h$ $=$ $\ds \int \paren {\lambda f + g} \overline h \rd \mu$ $\ds$ $=$ $\ds \int \paren {\lambda f \, \overline h} + \paren {g \, \overline h} \rd \mu$ $\ds$ $=$ $\ds \int \lambda f \, \overline h \rd \mu + \int g \, \overline h \rd \mu$ $\ds$ $=$ $\ds \lambda \int f \, \overline h \rd \mu + \int g \, \overline h \rd \mu$ $\ds$ $=$ $\ds \lambda \innerprod f h + \innerprod g h$

$\Box$

### $(3)$ Non-Negative Definiteness

 $\ds \innerprod f f$ $=$ $\ds \int f \, \overline f \rd \mu$ $\ds$ $=$ $\ds \int \cmod f^2 \rd \mu$ $\ds$ $\in$ $\ds \R_{\ge 0}$ Integral of Positive Function is Positive

$\Box$

### $(4)$ Positivity

Suppose that $\innerprod f f = 0$.

That is:

$\ds \int \cmod f^2 \rd \mu = 0$

Hence:

$\ds \int \cmod {f - 0}^2 \rd \mu = 0$

which is to say that $f = 0$ in $\map {L^2} \mu$ by definition of Lebesgue space.

$\Box$

Having verified all the axioms, we conclude $\innerprod \cdot \cdot$ is an inner product.

$\blacksquare$