The Inner Product on Rn and Cn

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 examine a familiar inner product on $\mathbb{R}^n$ ($\mathbb{C}^n$) - the dot product.

Definition: The Dot Product or Euclidean Inner Product on $\mathbb{R}^n$ (or $\mathbb{C}^n$) is defined for all $\vec{x} = (x_1, x_2, ..., x_n), \vec{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by $\displaystyle{\langle \vec{x}, \vec{y} \rangle = \sum_{k=1}^{n} x_ky_k}$.

For example:

(1)

\begin{align} \quad \langle (1, 3, 5), (4, -2, 4) = (1)(4) + (3)(-2) + (5)(4) = 4 - 6 + 20 = 18 \end{align}

It is easy to verify that the dot product is indeed an inner product on $\mathbb{R}^n$ and so $\mathbb{R}^n$ with the dot product is an inner product space.