I don't understand why we have <<f0, u>> instead of ||f0*u||^2 at "expanding 1st term" step. (or why we need to use the << >> notation)
Max
<<>> is the inner product. When expanding the term, we're doing this: ||f+eu||^2 = <<f+eu,f+eu>> = <<f,f>> + 2e<<f,u>> + e^2<<u,u>> = ||f||^2 + e^2||u||^2 + 2e<<f,u>>. This works out because an inner product is linear on both terms.
I think the slides use <<>> instead of <> for inner product here because we are working in a vector space of functions, but that's not really a notational rule.
I don't understand why we have <<f0, u>> instead of ||f0*u||^2 at "expanding 1st term" step. (or why we need to use the << >> notation)
<<>> is the inner product. When expanding the term, we're doing this: ||f+eu||^2 = <<f+eu,f+eu>> = <<f,f>> + 2e<<f,u>> + e^2<<u,u>> = ||f||^2 + e^2||u||^2 + 2e<<f,u>>. This works out because an inner product is linear on both terms.
I think the slides use <<>> instead of <> for inner product here because we are working in a vector space of functions, but that's not really a notational rule.