Standard Inner Products

Inner Product

Standard inner product of vector $u,v$ is: $(u,v)={\sum }_{k=1}^n{u}_i{v}_i=u_1{v}_1+\cdots +u_n{v}_n$

Length of Norm

The length of the normal $\mid \mid v\mid \mid =\sqrt{(v,v)}=\sqrt{v_1^2+\cdots +v_n^2}$ Note that the inner product is linear in the sense that $(u,av=bw)=a(u,v)+b(u,w)$

Properties of Standard Inner Product over ${\mathbb{R}}^n$

1. $(u,v)=(v,u)$
2. $(u,v_1+v_2)=(u,v_1)+(u,v_2)$
3. $(u,av)=a(v,u)$
4. $(u,u)\ge 0$ with equality if $u=\overrightarrow{0}$

Proofs

• Proving Standard Inner Function Property 2
• Proving Standard Inner Function Property 3