# A property of the canonical dual frame in a Hilbert space

Let $$\{ g_n \}$$ be a frame in a separable Hilbert space $$H$$. Then the frame operator $$S:H\to H$$ defined as $$$$S f := \sum_{n=1}^\infty (f,g_n)g_n$$$$ is a Hilbert space isomorphism i.e., continuous and bijective. Then one can define the canonica dual frame of $$g_n$$ as the frame $$S^{-1}g_n=:f_n$$. It is quite straightforward to prove that this is in fact a frame. Where I am having trouble is the following claim from the artile "Revisiting Landauʼs density theorems for Paley–Wiener spaces" in page 5 the authors claim that furthermore $$|\langle f_n,g_n \rangle | \leq 1$$. The reference that they give does not give a proof of this claim. A proof is given in Lemma 6.14 of "What is variable bandwidth", but to me this (two lines) proof is not transparent.

I still think that the claim holds, but I was not able to prove it.

I assume your inner product is conjugate linear in the second factor, based on your assertion that the operator $$S$$ is a Hilbert space iso. Then $$\langle f_n, g_n\rangle = \langle f_n, S f_n \rangle = \langle f_n, \sum_k \langle f_n, g_k\rangle g_k\rangle = \sum_k |\langle f_n, g_k\rangle|^2$$ is real and non-negative. Subtract $$|\langle f_n,g_n\rangle|^2$$ from both sides, you find $$|\langle f_n,g_n\rangle| - |\langle f_n,g_n\rangle|^2\geq 0$$ which can only hold if $$|\langle f_n,g_n\rangle| \leq 1$$.