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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×VK·,·:V×VK such that for all x,y,zV,αKx,y,zV,αK

  • 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,x0x,x0 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*xx,y:=i=1Nxiyi¯=y*x
  • V=C[a,b]V=C[a,b], x,y:=abx(t)y(t)¯dtx,y:=abx(t)y(t)¯dt

Note that a valid inner product space induces a normed vector space with norm x=x,xx=x,x. (Proof relies on Cauchy-Schwartz inequality.) In RNRN or CNCN, the standard inner product induces the 22-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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