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Let

$V$ be a vector space. We say that the inner product of vectors

$\vec{v_1}$ and

$\vec{v_2}$ , written

$<v_1,v_2>$ , is a function that takes any two vectors to a real number and has the following additional properties:
1. (Symmetry or Conjugate Symmetry) When dealing with only real numbers, the following symmetry formula must hold:

$<x,y> = <y,x>,$ and when dealing with complex numbers, the following conjugate symmetry formula holds:

$<x,y> = \overline{<y,x>},$ where

$\overline{<y,x>}$ denotes complex conjugation.

2. (Linearity in the first argument) The following formula holds for all scalars

$\alpha,\beta$ and vectors

$\vec{x},\vec{y},\vec{z}$ :

$<\alpha \vec{x}+\beta \vec{y},\vec{z}> = \alpha<\vec{x},\vec{z}> + \beta<\vec{y},\vec{z}>.$ Note in the case of real inner products we can factor out of the second term as

$<\vec{x},\alpha\vec{y}>=\alpha<\vec{x},\vec{y}>$ but if we are dealing with a complex inner product, factoring out of the second term results in a conjugate factor:

$<\vec{x},\alpha \vec{y}> = \overline{\alpha}<\vec{x},\vec{y}>.$ 3. (Positive definiteness) It is always true that

$< \vec{x},\vec{x} > \geq 0$ and

$<\vec{x},\vec{x}>=0$ if and only if

$\vec{x}=\vec{0}$ .

If

$V$ is a vector space and

$<\cdot,\cdot>$ is an inner product on

$V$ , then we say that

$H=(V,<\cdot,\cdot>)$ is an inner product space.

Examples of inner product spaces
1. Let

$\vec{x},\vec{y} \in \mathbb{R}^{n \times 1}$ with

$\vec{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$ and

$\vec{y}=\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}$ . Then the following formula defines an inner product:

$< \vec{x}, \vec{y} > = x_1y_1 + x_2y_2 + \ldots + x_ny_n = \displaystyle\sum_{k=1}^n x_ky_k.$ 2. Let

$\vec{p},\vec{q} \in \mathbb{P}$ (the vector space of polynomials). The following formula defines an inner product:

$<\vec{p},\vec{q}> = \displaystyle\int_0^{\infty} \vec{p}(x)\vec{q}(x)e^{-x} dx.$ 5. Let

$f,g \in C[0,1]$ , the functions continuous on the interval

$[0,1]$ . The following formula defines an inner product:

$<f,g>=\displaystyle\int_0^1 f(x)g(x)dx.$ 4. Let

$\{a_k\}_{k=0}^{\infty}, \{b_k\}_{k=0}^{\infty} \in \ell^1(\mathbb{R})$ , the set of sequences

$\{x_k\}$ such that

$\displaystyle\sum_{k=0}^{\infty} |x_k| < \infty$ . The following formula defines an inner product:

$<\{a_k\},\{b_k\}> = \displaystyle\sum_{k=0}^{\infty} a_kb_k.$ 5. Let

$z_1, z_2$ be complex numbers with

$z_1=a+bi$ and

$z_2=c+di$ and

$i^2=-1$ . The following formula defines an inner product:

$<z_1,z_2> = z_1 \overline{z_2},$ where

$\overline{z_2}=c-di$ (called the complex conjugate of

$z_2$ ).