Inner product

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In linear algebra, an inner product in a vector space is a function from to satisfying the following axioms for all vectors :[1]

  • , with if and only if ,
  • (the inner product is commutative),
  • , and
  • for all , .

One consequence of the inner product axioms is that the inner product is multilinear in both variables; that is:

The dot product in the Euclidean vector space is the best-known example of an inner product.

An inner product space is a vector space together with an inner product.


  1. Anton, Howard and Chris Rorres. Elementary Linear Algebra: Applications Version. 9th ed. N.p.:John Wiley & Sons, Inc., 2005. p. 296