Today I Learned

Some of the things I've learned every day since Oct 10, 2016

8: Inner Product Spaces

An inner product space is a vector space V over a field F along with the additional structure of a mapping called an inner product, a generalization of the dot product.

The inner product \langle \cdot , \cdot \rangle : V \times V \rightarrow F must satisfy the following properties:

linearity in the first argument

\forall a \in F, x, y \in V:

\langle a x, y \rangle = a \langle x, y \rangle

\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle

positive-definiteness

\langle x, x \rangle \geq 0

\langle x, x \rangle = 0 \Longleftrightarrow x = \boldsymbol{0}

conjugate symmetry

\langle x, y \rangle = \overline{ \langle y, x \rangle}

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