Inner Product Spaces Over the Field of Real Numbers

Inner Product Spaces Over the Field of Real Numbers

We will soon show that the set of all square Lebesgue integrable functions is an inner product space, but of course, we will first need to formally define an inner product space and inner product (which the reader is likely already familiar with).

Definition: Let $V$ be a vector space over the field $\mathbb{R}$. A two variable function $(\cdot, \cdot ): V \times V \to \mathbb{R}$ is called an Inner Product on $V$ if it satisfies the following properties:
1) $(x, x) \geq 0$ for all $x \in V$, and $(x, x) = 0$ if and only if $x = 0$.
2) $(x + y, z) = (x, z) + (y, z)$ for all $x, y, z \in V$.
3) $(\alpha x, y) = \alpha (x, y)$ for all $x, y \in V$ and for all $\alpha \in \mathbb{R}$.
4) $(x, y) = (y, x)$ for all $x, y \in V$.

Note that we $(x, y)$ is allowed to equal to $0$ if $x \neq y$ and $x, y \neq 0$. We only require that the inner product of an element with ITSELF if equal to $0$ if and only if that element is $0$.

Of course, inner products can be defined on vector spaces over other fields such as the set of complex numbers, $\mathbb{C}$.

Definition: If $V$ is a vector space over the field $\mathbb{R}$ and $(\cdot, \cdot)$ is an inner product of $V$ then $(V, (\cdot, \cdot))$ is said to be an Inner Product Space.

Perhaps the most familiar inner product space for the reader is the set $\mathbb{R}^n$ with the inner product $(\cdot, \cdot)$ defined for all vectors $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by:

\begin{align} (\mathbf{x}, \mathbf{y}) = x_1y_1 + x_2y_2 + ... + x_ny_n = \sum_{k=1}^{n} x_ky_k \end{align}

This inner product is common denoted $\mathbf{x} \cdot \mathbf{y}$ as opposed to the notation $(\mathbf{x}, \mathbf{y})$. If we consider the $\mathbb{R}^3$ and the vectors $\mathbf{x} = (1, 2, 3)$ and $\mathbf{y} = (2, -1, 4)$ then the inner product between these two vectors is:

\begin{align} \quad \mathbf{x} \cdot \mathbf{y} = (1, 2, 3) \cdot (2, -1, 4) = (1)(2) + (2)(-1) + (3)(4) = 12 \end{align}

It is very easy to verify that this is indeed an inner product on $\mathbb{R}^n$ and it is given a special name - the Euclidean inner product on $\mathbb{R}^n$ or simply, the dot product.

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