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# Inner product

An inner product satisfies the following properties for two vectors $$v$$ and $$w$$, and a scalar $$\alpha$$:

1. $$\langle u+v, w\rangle=\langle u,w\rangle+\langle v,w\rangle$$,
2. $$\langle\alpha v,w\rangle=\alpha\langle v,w\rangle$$,
3. $$\langle v,w\rangle=\langle w,v\rangle$$, and
4. $$\langle v,v\rangle\geq 0$$ and $$\langle v,v\rangle=0$$ if and only if $$v=0$$.