Inner Products and Inner Product Spaces

Definition: Let $X$ be a linear space. A function $\langle \cdot, \cdot \rangle : X \times X \to \mathbb{C}$ is an Inner Product on $X$ if the following properties are satisfied for all $x, y, z \in X$ and for all $\lambda \in \mathbb{C}$: 1) $\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0$ if and only if $x = 0$. (Positive Definiteness Property) 2) $\langle x, y \rangle = \overline{\langle y, x \rangle}$. (Conjugate Symmetry Property) 3) $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$ and $\langle \lambda x, y \rangle = \lambda \langle x, y \rangle$. (Linearity in the First Component Property)

If $X$ is a linear space over $\mathbb{R}$ then the second condition is replaced with $\langle x, y \rangle = \langle y, x \rangle$ for all $x, y \in X$.

There is a similar linearity property for the second component of inner products which is proven in the proposition below.

Proposition 1: Let $X$ be a linear space and let $\langle \cdot, \cdot \rangle$ be an inner product on $X$. Then for all $x, y, z \in X$ and for all $\lambda \in \mathbb{C}$ we have that: a) $\langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle$. b) $\langle x, \lambda y \rangle = \overline{\lambda} \langle x, y \rangle$.

If $X$ is a linear space over $\mathbb{R}$ the (b) is replaced with $\langle x, \lambda y \rangle = \lambda \langle x, y \rangle$ for all $\lambda \in \mathbb{R}$.

• Proof of a): Let $x, y, z \in X$. Then:

(1)

\begin{align} \langle x, y + z \rangle &= \overline{\langle y + z, x \rangle} \\ &= \overline{\langle y, x \rangle + \langle z, x \rangle} \\ &= \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} \\ &= \langle x, y \rangle + \langle x, z \rangle \quad \blacksquare \end{align}

• Proof of b) Let $x, y \in X$ and let $\lambda \in \mathbb{C}$. Then:

(2)

\begin{align} \quad \langle x , \lambda y \rangle &= \overline{\langle \lambda y, x \rangle} \\ &= \overline{\lambda \langle y, x \rangle} \\ &= \overline{\lambda} \overline{ \langle y, x \rangle} \\ &= \overline{\lambda} \langle x, y \rangle \quad \blacksquare \end{align}

Definition: An Inner Product Space is a linear space $H$ paired with an inner product $\langle \cdot, \cdot \rangle : H \times H \to \mathbb{C}$ on $H$.

Later we will see that every inner product space is a normed space with the norm defined on $H$ to be $\| x \| = \langle x, x \rangle^{1/2}$.

One of the simplest examples of an inner product space is $\mathbb{R}^n$ with the usual Euclidean dot product defined for all $x = (x_1, x_2, ..., x_n), y = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by:

(3)

\begin{align} \quad \langle x, y \rangle = x_1y_1 + x_2y_2 + ... + x_ny_n \end{align}

And similarly, $\mathbb{C}^n$ with the usual Euclidean dot product defined for all $x = (x_1, x_2, ..., x_n), y = (y_1, y_2, ..., y_n) \in \mathbb{C}^n$ by:

(4)

\begin{align} \quad \langle x, y \rangle = x_1\overline{y_1} + x_2\overline{y_2} + ... + x_n \overline{y_n} \end{align}

For another example, let $(X, \mathfrak T, \mu)$ be a measure space. Then the Lebesgue space $L^2(X, \mathfrak T, \mu)$ is an inner product space with inner product defined for all $f, g \in L^2(X, \mathfrak T, \mu)$ by:

(5)

\begin{align} \quad \langle f, g \rangle = \int_X f \overline{g} \: d \mu \end{align}