Vector space

Inner product space

An inner product space is a vector space (𝑉,𝕂)¹ together with an operation

⟨⋅,⋅⟩:𝑉×𝑉→𝕂

which for any 𝑥,𝑦,𝑧 ∈𝑉 and 𝜆,𝜇 ∈𝕂 has the following properties: vec

1. conjugate symmetry ⟨𝑥,𝑦⟩ =――――⟨𝑦,𝑥⟩
2. linearity in the first argument^[Alternatively in the second, see below] ⟨𝜆𝑥 +𝜇𝑦,𝑧⟩ =𝜆⟨𝑥,𝑧⟩ +𝜇⟨𝑦,𝑧⟩
3. positive-definiteness ⟨𝑥,𝑥⟩ >0 ⟺ 𝑥 ≠𝟎

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 ⟨𝑥|𝜆𝑦 +𝜇𝑧⟩ =𝜆⟨𝑥|𝑦⟩ +𝜇⟨𝑥|𝑧⟩

Every inner product induced a norm ‖𝑥‖2 =⟨𝑥,𝑥⟩, and norms have a corresponding unique inner product iff the Parallelogram law holds.

Properties

1. antilineärity in the other argument: ⟨𝜆𝑥+𝜇𝑦|𝑧⟩ =𝜆∗⟨𝑥|𝑧⟩ +𝜇∗⟨𝑦|𝑧⟩
2. general Cauchy-Schwarz inequality: |⟨𝑥|𝑦⟩|2 ≤‖𝑥‖2‖𝑦‖2

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 𝕂 =ℂ or 𝕂 =ℝ, in the latter case conjugate symmetry (1) is just symmetry ↩

2. See Vector notation in these notes ↩