NAVIGATION

Home

Research

Bookshelf

Garden

FIND ME ON

GitHub

LinkedIn

Email

Home

Research

Bookshelf

Garden

Inner Product

Definition (Inner product)

An inner product of an F\mathbb{F}-Vector Space VV assigns to vectors v1,v2∈Vv_{1},v_{2}\in V the number ⟨v1,v2⟩∈F\langle v_{1}, v_{2} \rangle\in\mathbb{F} and the assignment satisfies the following rules:

  1. Symmetry⟨v1,v2⟩=⟨v2,v1⟩‾\langle v_{1}, v_{2} \rangle =\overline{\langle v_{2}, v_{1} \rangle} for v1,v2∈Vv_{1},v_{2}\in V.
  2. Linearity⟨a1v1+a2v2,v⟩=a1⟨v1,v⟩+a2⟨v2,v⟩\langle a_{1}v_{1}+a_{2}v_{2}, v \rangle=a_{1}\langle v_{1}, v \rangle +a_{2}\langle v_{2}, v \rangle for a1,a2∈Fa_{1},a_{2}\in\mathbb{F}
  3. Positivity ⟨v,v⟩≥0\langle v, v \rangle \ge 0for v∈Vv\in V
  4. Definiteness ⟨v,v⟩=0  ⟺  v=0v\langle v, v \rangle =0\iff v=0_{v}

Proposition (Parallelogram law)

Let VV be an Inner Product Space. If v,w∈Vv,w\in V, then ∥v+w∥2+∥v−w∥2=2(∥v∥2+∥w∥2)\lVert v+w \rVert ^{2}+\lVert v-w \rVert ^{2}=2(\lVert v \rVert ^{2}+\lVert w \rVert ^{2})

Theorem (2.2.1)

For an F\mathbb{F}-Inner Product Space (V,⟨⋅,⋅⟩)(V,\langle \cdot, \cdot \rangle) we have ∣⟨x,y⟩∣≤<x,x><y,y>x,y∈V| \langle x,y \rangle |\le \sqrt{ \left< x,x \right> }\sqrt{ \left< y,y \right> }\quad x,y\in V with equality if and only if x=αy,α∈Rx=\alpha y,\alpha \in \mathbb{R}.

Theorem (2.2.2)

If xn→xx_{n}\to x and yn→yy_{n}\to y, then <xn,yn>→<x,y>\left< x_{n},y_{n} \right>\to \left< x,y \right>.

Linked from

Hilbert Space

Inner Product Space

L2 space

Riesz representation theorem