Notes on the Harmonic Oscillator and the Fourier Transform

Square-integrable functions f ∈ L are those of length ‖f‖L2 = 〈f, f〉 L2 < ∞. A subset D ⊂ L is said to be dense, if any f ∈ L can be approximated by a sequence fn ∈ D. This means limn ‖f − fn‖L2 = 0. An orthonormal basis {Ωn} is a set of orthonormal vectors whose finite linear combinations are dense. A linear transformation T is continuous on L if ‖Tf‖L2 ≤ M‖f‖L2 for some constant M < ∞. A continuous transformation defined on a basis extends uniquely to all L.