Inner product

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In linear algebra, an inner product \langle \cdot, \cdot \rangle in a vector space V is a function from V \times V to \mathbb{R} satisfying the following axioms for all vectors \vec{u}, \vec{v}, \vec{w} \in V:[1]

  • \langle \vec{v}, \vec{v}\rangle \geq 0, with \langle \vec{v}, \vec{v}\rangle = 0 if and only if \vec{v} = \vec{0},
  • \langle \vec{v}, \vec{w}\rangle = \langle \vec{w}, \vec{v} \rangle (the inner product is commutative),
  • \langle \vec{u} + \vec{v}, \vec{w} \rangle = \langle \vec{u}, \vec{w} \rangle + \langle \vec{v}, \vec{w} \rangle, and
  • for all k \in \mathbb{R}, \langle k\vec{v}, \vec{w}\rangle = k\langle \vec{v}, \vec{w}\rangle.

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

  • \langle \alpha \vec{u} + \beta \vec{v}, \vec{w}\rangle = \alpha \langle \vec{u}, \vec{w} \rangle + \beta \langle \vec{v}, \vec{w}\rangle
  • \langle \vec{u}, \alpha \vec{v} + \beta \vec{w}\rangle = \alpha \langle \vec{u}, \vec{v} \rangle + \beta \langle \vec{u}, \vec{w}\rangle

The dot product in the Euclidean vector space \mathbb{R}^n 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
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