For x in H, That gives you one of the two inequalities that you need for a frame. The other one comes from the Banach inversion theorem, which says that T (and therefore also T*) must have a bounded inverse. It follows that and hence
Hi! I came across this problem while I was reading a supplementary material for our course on hilbert spaces. Can anyone help me with this? Thank you!
given an orthonormal basis { } of a hilbert space H and a one-to-one and onto continuous linear transformation T from H to H, prove that {T } is a frame for H.
For x in H, That gives you one of the two inequalities that you need for a frame. The other one comes from the Banach inversion theorem, which says that T (and therefore also T*) must have a bounded inverse. It follows that and hence