Vector space

Inner product space

An inner product space is a vector space ¹ together with an operation

which for any and has the following properties: vec

1. conjugate symmetry
2. linearity in the first argument^[Alternatively in the second, see below]
3. positive-definiteness

In some fields a bra-ket notation style inner product is more common, signaled by a | instead of ,², in which case the second axiom is

2. linearity in the second argument

Every inner product induced a norm , and norms have a corresponding unique inner product iff the Parallelogram law holds.

Properties

1. antilineärity in the other argument:
2. general Cauchy-Schwarz inequality:

Further properties

• A change of basis is also a Change of inner product
• The inner product is continuous

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tidy | en | sembr

Footnotes

1. Where either or , in the latter case conjugate symmetry (1) is just symmetry ↩

2. See Vector notation in these notes ↩