inner product

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Inner Product

An inner product on a vector space V over field K is a function ⟨,⟩:V×V→K such that for all x,y,z∈V and all c∈K we have

  1. ⟨x,x⟩ is real, nonnegative. Also, ⟨x,x⟩=0⟺x=0
  2. ⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩
  3. ⟨cx,z⟩=c⟨x,z⟩ (2 and 3 indicate that this is linear)
  4. ⟨x,z⟩=⟨z,x⟩¯ (ie symmetric)

If all of these hold, then V is an inner product space.

Example

Cn over C.

  • Euclidean inner product: ⟨x,z⟩:=z∗x
  • some A∈Mn positive definite: ⟨x,z⟩A:=z∗Ax
    Note that the euclidean inner product is a special case of the second, where A is the identity!

Mentions

Mentions

File Last Modified
2025-01-27 graphs lecture 2 2025-02-12
2025-04-01 equivariant lecture 9 2025-04-01
additivity of inner products 2025-08-17
cauchy-schwarz theorem 2025-08-17
Functional Analysis Lecture 13 2025-07-25
inner products define norms 2025-08-17
interpretation of the symmetric graph laplacian 2025-08-17
Lecture 22 2025-08-17
Lecture 23 2025-08-17
Lecture 36 2025-08-17
self-adjoint 2025-03-31

{ .block-language-dataview}
Created 2025-09-04 ֍ Last Modified 2025-09-09