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Inner Product MOC

  • Inner product is a generalization of the euclidean dot product
  • It is denoted by $\langle\cdot,\cdot\rangle$
  • It has to satisfy certain properties but I don’t think I will go into what those are
  • It gives intuitive sense of lengths, angles, etc. for any vector space

Important details

Every inner product induces a valid norm

If you have some inner product $\langle\cdot,\cdot\rangle$ then you can define a norm as follows

$$ ||u|| = \sqrt{\langle u, u\rangle} $$

Inner product spaces are a proper subset of normed vector spaces

  • Every inner product space has a valid norm induced by it.
  • Not every normed vector spaces have a valid inner product.