The Inner Product on ℓ2 and L2(E)

Recall from the Inner Products and Inner Product Spaces page that an inner product space is a linear space $X$ with function $\langle \cdot, \cdot \rangle : X \times X \to \mathbb{C}$ (or $\mathbb{R}$) such that:

• 1. $\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0$ if and only if $x = 0$.

• 2. $\langle x, y \rangle = \overline{\langle y, x \rangle}$ for all $x, y \in X$.

• 3. $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$ and $\langle \lambda x, y \rangle = \lambda \langle x, y \rangle$ for all $x, y, z \in X$ and all $\lambda \in \mathbb{C}$

We will now define an inner product on $\ell^2$ and one on $L^2(E)$.

An Inner Product on ℓ2

Definition: We define the following inner product on $\ell^2$ for all sequences $(x_n), (y_n) \in \ell^2$ by $\displaystyle{\langle (x_n), (y_n) \rangle = \sum_{n=1}^{\infty} x_ny_n}$.

An Inner Product on L^2(E)

Definition: We define the following inner product on $L^2(E)$ for all $f, g \in L^2(E)$ by $\displaystyle{\langle f, g \rangle = \int_E f(x)g(x) \: dx}$.