§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(-¥, ¥).