Inner Product and Inner Product Space

Introduction

We have learned real inner product very well in high school or college. However, when it comes to sophisticated inner product and inner product space, such as complex inner product and complex inner product space, sometimes it is confusing.

In this blog post, I would like to discuss the inner product, inner product space, and a definition discrepancy that sometimes people ignored.

Inner Product and Inner Product Space

Vector Space

In case someone does not know what vector space, it was defined clearly in Wikipedia and it was not hard to understand. The only caveat is probably that in addition to the conventional valued vector space, function space is also a vector space.

Inner Product

An inner product (also called a dot product or scalar product) on a complex vector space $\mathbb{V}$ is a function

$$
\langle -, - \rangle : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{C}
$$

that satisfies the following four properties:

1. $\langle v_1 + v_2, w \rangle = \langle v_1, w \rangle + \langle v_2, w \rangle$ for all $v_1, v_2, w \in \mathbb{V}$
2. $\langle cv, w \rangle = c \langle v, w \rangle$ for all $c \in \mathbb{C}$ and all $v, w \in \mathbb{V}$
3. $\langle w, v \rangle = \overline{ \langle v, w \rangle }$ for all $v, w \in \mathbb{V}$
4. $\langle v, v \rangle$ is a positive real number for any $v \in \mathbb{V} \sim \{0\}$

Note that there is discrepancy on the property 2. Some people defined the property 2 to be $\langle cv, w \rangle = \overline{c}\langle v, w \rangle$ for all $c \in \mathbb{C}$ and all $v, w \in \mathbb{V}$. The difference would affect the definition of the inner product space. Many people were not aware of this definition difference and made mistakes sometimes. For inner product of real numbers, there is no such discrepancy since $c = \overline{c}$ for any $c \in \mathbb{R}$.

Derived Inner Product Properties

Based on the four inner product properties, we could further derive more inner product properties.

$\langle v, w_1 + w_2 \rangle = \langle v, w_1 \rangle + \langle v, w_2 \rangle$ for all $v, w_1, w_2 \in \mathbb{V}$

This is because

$$
\begin{align}
\langle v, w_1 + w_2 \rangle &= \overline{\langle w_1 + w_2, v \rangle}\\
&= \overline{\langle w_1, v \rangle + \langle w_2, v \rangle} \\
&= \overline{\langle w_1, v \rangle} + \overline{\langle w_2, v \rangle} \\
&= \langle v, w_1 \rangle + \langle v, w_2 \rangle
\end{align}
$$

$\langle v, cw \rangle = \overline{c} \langle v, w \rangle$ for all $c \in \mathbb{C}$ and all $v, w \in \mathbb{V}$

This is because

$$
\begin{align}
\langle v, cw \rangle &= \overline{\langle cw, v \rangle} \\
&= \overline{c \langle w, v \rangle} \\
&= \overline{c} \overline{\langle w, v \rangle} \\
&= \overline{c} \langle v, w \rangle \\
\end{align}
$$

Note that $\overline{ab} = \overline{a}\overline{b}$ and this could be easily verified.

Inner Product Space

An inner product space is a vector space along with an inner product.

This sounds awkward. But it is just literally a vector space with an inner product expression defined.

For example, we have the following complex inner product space.

Given vector space $\mathbb{C}^n$, we have two vectors $v = \{v_1, v_2, \cdots, v_n\}, w = \{w_1, w_2, \cdots, w_n\} \in \mathbb{C}^n$, the inner product is defined as

$$
\langle v, w \rangle = \sum_{i=1}^{n} v_i \overline{w}_i
$$

Note that the above inner product definition was defined given the property 2 of inner product was defined as $\langle cv, w \rangle = c \langle v, w \rangle$.

If the the property 2 of inner product was defined as $\langle cv, w \rangle = \overline{c}\langle v, w \rangle$, then the above inner product definition would become

$$
\langle v, w \rangle = \sum_{i=1}^{n} \overline{v}_i w_i
$$

This discrepancy also extends to other complex inner product spaces such as for $\mathbb{C}^{m \times n}$, etc.

We would see, probably in my future blogs, that we have such different inner product definitions under different inner product properties is to satisfy some special properties of Hermitian matrix, so that some mathematical conclusions would be unified regardless of which way you defined the property 2 of inner product.

Conclusions

Always pay close attention to the definitions in mathematics.