Hilbert Spaces
§Hilbert Space: a complete inner product space. (We shall not discuss the completeness here.)
§
• Examples:
•
vCn  with the usual inner product.
vCn ×m with the following inner product 
• <A, B> := Trace A*B         " A, B Î Cn ×m 
vL2[a,b]: all square integrable and Lebesgue measurable functions defined on an interval [a,b] with the inner product  
v                                           
•                <f, g> := aòb f(t)*g(t)dt,   Matrix form: <f, g> := aòb Trace[ f(t)*g(t)]dt. 
•                                                
vL2 = L2 (-¥, ¥): <f, g> := - ¥ ò¥ Trace[ f(t)*g(t)]dt.
v
v L2+ =  L2[0, ¥): subspace of  L2(-¥, ¥).
v L2- =  L2(-¥, 0]: subspace of L2(-¥, ¥).