Elementary Identities in a Real Inner Product Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and zero vector $0_{E}$; for $x,y\in E$, $-x$ is the additive inverse of $x$ and $x-y=x+(-y)$ as in Elementary Identities in a Vector Space, and for $\lambda\in\mathbb{R}$, $|\lambda|$ is its absolute value. Then for all $x,y,z\in E$ and all $\lambda\in\mathbb{R}$ the following hold.

1. (Bilinearity)¶ $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$, $\langle x,\lambda y\rangle=\lambda\langle x,y\rangle$, $\langle x-y,z\rangle=\langle x,z\rangle-\langle y,z\rangle$, $\langle x,y-z\rangle=\langle x,y\rangle-\langle x,z\rangle$, and $\langle -x,y\rangle=\langle x,-y\rangle=-\langle x,y\rangle$.

2. (Zero vector)¶ $\langle x,0_{E}\rangle=\langle 0_{E},x\rangle=0$ and $|0_{E}|=0$.

3. (Vanishing)¶ $|x|=0$ if and only if $x=0_{E}$; consequently $0<|x|$ whenever $x\ne 0_{E}$.

4. (Homogeneity)¶ $|\lambda x|=|\lambda|\,|x|$; in particular $|-x|=|x|$.

5. (Expansion)¶ $|x+y|^{2}=|x|^{2}+2\langle x,y\rangle+|y|^{2}$ and $|x-y|^{2}=|x|^{2}-2\langle x,y\rangle+|y|^{2}$.

6. (Parallelogram law)¶ $|x+y|^{2}+|x-y|^{2}=2|x|^{2}+2|y|^{2}$.

7. (Polarisation)¶ $4\langle x,y\rangle=|x+y|^{2}-|x-y|^{2}$.