Inner product of a Vector Space
An inner product on a vector space $V$ is a function $\langle.,.\rangle:V\times V\rightarrow \R$ satisfying the following:
- $\langle v,v\rangle ~> 0~~ \forall~~ v \in V~\backslash \{0\};$ $\langle v,v\rangle =0$ if and only if $v=0$
- $\langle v_1+v_2,v_3\rangle = \langle v_1,v_3 \rangle + \langle v_2,v_3 \rangle$
- $\langle v_1,v_2 \rangle = \langle v_2,v_1 \rangle$
- $\langle cv_1, v_2 \rangle = c\langle v_1,v_2 \rangle = \langle v_1,cv_2 \rangle$ , where $c \in \R$
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💡 A vector space $V$ together with an inner product $\langle.,.\rangle$ is called an inner product space.
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The Dot product is an example of Inner Product
Norm on a Vector Space
<aside>
💡 A norm on a vector space $V$ is a function
$||.||: V \rightarrow \R$
$x \rightarrow ||x||$
satisfying the following conditions:
- $||x+y|| \leq ||x|| + ||y||~, ~\forall x,y \in V$
- $||cx||= ~|c|||x||,
\forall~c \in \R$and for all $x\in V$
- $||x||\geq0~~\forall
x\in V ;||x||=0$ if and only if $x=0$
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Length as an example of a Norm
The Inner Product induces a Norm
