Inner product

  • Thread starter squenshl
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  • #1
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Homework Statement


Denote the inner product of f,g [itex]\in[/itex] H by <f,g> [itex]\in[/itex] R where H is some(real-valued) vector space
a) Explain linearity of the inner product with respect to f,g. Define orthogonality.
b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be defined as (give reasons)
i) <f,g> = integral from 0 to 1 of f(x)g(x) dx?
ii) <f,g> = integral from 0 to 1 [itex]\lambda[/itex](x)f'(x)g'(x) dx? where prime denotes the derivative and [itex]\lambda[/itex](x) > 0 is a smooth function (assuming f',g' [itex]\in[/itex] H)?
iii) <f,g> = f(x)g(x)?

Homework Equations





The Attempt at a Solution


a) That is just trivial
b) Not too sure on any of them
 

Answers and Replies

  • #2
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For i) since the integral is a real number (provided that f and g are continuous, differentiable etc.) once computed suggest that this will be an inner product, similarly for ii) and iii) (except the integral part for iii))
 
  • #3
HallsofIvy
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An inner product must be such that
[tex]<au+ bv, w>= a<u, w>+ b<v, w>[/tex]
[tex]<u, v>= \overline{<v, u>}[/tex]

Can you show that those are true for each of the given operations?
 
  • #4
479
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Do I show positive definiteness <f,f> >= 0? and <f,f> = 0 if and only if f = 0
 
  • #5
479
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I just proved all 3 axioms to be an inner product.
I got that all 3 operations are all defined as inner products, is this correct (I'm a little skeptical on ii))?
i) and iii) are easy.
 

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