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Inner Product Spaces

Module by: Mark A. Davenport. E-mail the author

Where normed vector spaces incorporate the concept of length into a vector space, inner product spaces incorporate the concept of angle.

Definition 1

Let VV be a vector space over KK. An inner product is a function 〈·,·〉:V×V→K〈·,·〉:V×V→K such that for all x,y,z∈V,α∈Kx,y,z∈V,α∈K

  • IP1. 〈 x , y 〉 = 〈 y , x 〉 ¯ 〈 x , y 〉 = 〈 y , x 〉 ¯
  • IP2. 〈 α x , y 〉 = α 〈 x , y 〉 〈 α x , y 〉 = α 〈 x , y 〉
  • IP3. 〈 x + y , z 〉 = 〈 x , z 〉 + 〈 y , z 〉 〈 x + y , z 〉 = 〈 x , z 〉 + 〈 y , z 〉
  • IP4. 〈x,x〉≥0〈x,x〉≥0 with equality iff x=0x=0.

A vector space together with an inner product is called an inner product space.

Example 1

  • V=CNV=CN, 〈x,y〉:=∑i=1Nxiyi¯=y*x〈x,y〉:=∑i=1Nxiyi¯=y*x
  • V=C[a,b]V=C[a,b], 〈x,y〉:=∫abx(t)y(t)¯dt〈x,y〉:=∫abx(t)y(t)¯dt

Note that a valid inner product space induces a normed vector space with norm x=〈x,x〉x=〈x,x〉. (Proof relies on Cauchy-Schwartz inequality.) In RNRN or CNCN, the standard inner product induces the ℓ2ℓ2-norm. We summarize the relationships between the various spaces introduced over the last few lectures in Figure 1.

Figure 1: Venn diagram illustrating the relationship between vector and metric spaces.
A Venn diagram illustrating the relationships between the various mathematical spaces discussed up to this point. Metric spaces and vector spaces have a nonzero intersection, and normed vector spaces lie within this intersection. Inner product spaces lie within the set of normed vector spaces.

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