inner product

Definition

Suppose a real vector space, with vectors $u,v,w$ and scalar $\alpha$, then an inner product $\langle \cdot, \cdot \rangle$ satisfies properties:

• bilinearity
  • $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$
  • $\langle \alpha v, w \rangle = \alpha \langle v,w \rangle$
• symmetry
  • $\langle v,w \rangle = \langle w,v \rangle$
• positive definiteness
  • $\langle v,v \rangle \geq 0$
  • $\langle v,v \rangle = 0$ iff $v = 0$

Notes

• generalization of dot product, where $\langle u, v \rangle = v'w = \sum_i v_i w_i$

References

1. https://mathworld.wolfram.com/InnerProduct.html
2. https://people.eecs.berkeley.edu/~bartlett/courses/281b-sp08/7.pdf