Real Inner Product Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $E$ be a vector space over $\mathbb{R}$, and let $0_{E}$ be its zero vector.

1. (Inner product)¶ An inner product on $E$ is a map assigning to each pair $x,y$ of elements of $E$ a real number $\langle x,y\rangle$, subject to the following conditions for all $x,y,z\in E$ and all $\lambda\in\mathbb{R}$.

(a) (Symmetry) $\langle x,y\rangle=\langle y,x\rangle$.

(b) (Additivity in the first argument) $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$.

(c) (Homogeneity in the first argument) $\langle \lambda x,y\rangle=\lambda\langle x,y\rangle$.

(d) (Positive definiteness) $0\le\langle x,x\rangle$, and $\langle x,x\rangle=0$ only if $x=0_{E}$.

A real inner product space is a vector space over $\mathbb{R}$ together with an inner product on it. For the remainder of this item, $E$ together with $\langle\cdot,\cdot\rangle$ is a real inner product space.

2. (Norm)¶ Let $x\in E$. Since $0\le\langle x,x\rangle$ by (d), Existence and Uniqueness of the Nonnegative Square Root provides a unique real number $r$ with $0\le r$ and $r^{2}=\langle x,x\rangle$. The norm of $x$ is this number, written $|x|$; thus $0\le|x|$ and $|x|^{2}=\langle x,x\rangle$.

3. (Distance)¶ For $x,y\in E$ the distance from $x$ to $y$ is the real number $d(x,y)=|x-y|$, where $x-y=x+(-y)$ is the difference of Elementary Identities in a Vector Space.

4. (Ambient notation)¶ When several real inner product spaces are in play, the inner product, norm and distance of $E$ are written $\langle\cdot,\cdot\rangle_{E}$, $|\cdot|_{E}$ and $d_{E}$. Conditions (b) and (c) place the linearity requirements on the first argument; this convention and the notation $|\cdot|$ for the norm differ from those of a complex inner product space, whose inner product is linear in its second argument.