Answer to Question #272578 in Functional Analysis for Prathibha Rose

Question

1.Show that an operator T on a Hilbert Space H is unitary iff

T(𝑒𝑖) complete orthonormal set whenever 𝑒𝑖

is.

Answer 1

Hilbert space H₁ has some orthonormal basis {e_i}_i∈I . Then since unitary operator T maps any orthonormal basis of H₁to an orthonormal basis of H₂, then specifically {T(ei)}_i∈I must be an orthonormal basis of H₂.