(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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)?

2. Relevant equations

3. The attempt at a solution

a) That is just trivial

b) Not too sure on any of them

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Inner product

**Physics Forums | Science Articles, Homework Help, Discussion**