orthogonal expansion(hilbert space)
these 4 are the porblems from the book "an introduction to hilbert space" by young.these are the problems from page 42.porblem 4.1,4.2,4.7,4.8 another one is 4.6.
1) Find a vector in ¢^3(complex) orthogonal to (1,1,1) and (1,ω,ω^2) where ω=e^(2πi/3)
2) Let e_j(z)=z^j for z belongs to ¢, j belongs to z(integer).show that (e_j) that ranges from (-)infinity to (+)infinity is an orthonormal sequence in RL^2.
3) the first three legendre polynomials are P_0(x)=1, P_1(x)=x P_2(x)=(3x^2-1)/2 .show that the orthonormal vectors in L^2(-1,1) obtained by applying the Gram-Schmid process to 1,x,x^2 are scalar multiples of these.
4) Which point In the linear span of (1,ω,ω^2) and (1,ω^2,ω) where ω=e^(2πi/3), is nearest to (1,-1,1)?
These are the problems of Orthogonal expansion of Hilbert space.can anybody send me those solutions?