(a) Suppose dim H = d, and also suppose that U satisfies U†U = 1. Pick an orthonormal basis {|k}, and show that the set {U |k} is also an orthonormal basis. Use this fact to prove that UU† = 1 also,...

(a) Suppose dim H = d, and also suppose that U satisfies U†U = 1. Pick an orthonormal basis {|k}, and show that the set {U |k} is also an orthonormal basis. Use this fact to prove that UU† = 1 also, so that U must be unitary.

(b) Now imagine an infinite dimensional Hilbert space, with an infinite orthonormal basis set {|1, |2, |3, ...}. Define the operator U by

Show that U†U = 1, but UU† = 1. Thus U is not unitary.