Question

In C[-pi, pi] , consider the set {1/Sqrt[2], cos(2x), sin(2x)} and define the inner product as =1/Pi \[\int_{a}^{b} f(x) g(x) dx\]

Answer 1

a) find ||1/sqrt[2]||

Answer 2

@Zarkon

Answer 3

@satellite73

Answer 4

but \[\frac{1}{\sqrt{2}}\] is a number right, so you are treating it as a constant function

Answer 5

reminds me of the fourier stuff ive been looking at

Answer 6

orthogonal functions and what not ....

Answer 7

i take it you mean inner product is
\[<f,g>=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)g(x)dx\]

Answer 8

not the inner product equals \(\frac{1}{\pi}\) and not the integral from some a to some b

Answer 9

yes,

Answer 10

ok so replace both \(f\) and \(g\) by \(\frac{1}{\sqrt{2}}\)

Answer 11

because \(||f||=<f,f>\)

Answer 12

\[\int_{-pi}^{pi} 1/2 dx\]

Answer 13

i made a mistake, norm is root of that

Answer 14

yea, that is right
\[\frac{1}{\pi}\int_{-\pi}^{\pi}\frac{1}{2}dx\]

Answer 15

i think you get 1

Answer 16

actually i am sure you get 1

Answer 17

\[\frac{1}{\pi}\times 2\pi\times \frac{1}{2}=1\]

Answer 18

and \(\sqrt{1}=1\)

Answer 19

nice , thanks

Answer 20

yw, my pleasure

Answer 21

each of these has norm 1 i think