A sequence of distinct vectors $\{g_1,g_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that $$A\|g\|^2\leq\sum_{n=1}^\infty |(g,g_n)|^2\leq B \|g\|^2.$$ for each $g\in H$. Prove that $\{g_n\}$ is a frame if and only if the so called frame operator $$Sg=\sum_n (g,g_n) g_n$$ is continuous and continuously invertible on its range.

Any suggestions to approach a proof, please?

  • $\begingroup$ What is $g$ in your question? $\endgroup$ Jun 9, 2015 at 8:51
  • $\begingroup$ An element of H. I've edited my post. $\endgroup$
    – Mark
    Jun 9, 2015 at 8:52

1 Answer 1


For general normed linear spaces $X,Y$ there is the following characterisation of normed space isomorphisms:

The operator $T : X \to Y$ is a normed space isomorphism if, and only if, there exists constants $A,B > 0$ such that, for all $x \in X$, $$ A \| x \|_X \leq \| T x \|_Y \leq B\| x \|_X. $$

Note that a normed space isomorphism is, by definition, a continuous mapping with continuous inverse on its range. Do you see how to proceed from here?

EDIT: Here are, as requested, some steps:

  • Note that the frame inequality can be written in terms of the frame operator as $$ A \|x\|^2_{\mathcal{H}} \leq \langle S x, x \rangle_{\mathcal{H}} \leq B \|x\|^2_{\mathcal{H}}. $$

  • The frame operator $S$ is positive, and hence an application of Cauchy-Schwarz gives $$ \langle Sx , x \rangle \leq \| Sx \| \|x\|. $$

Do not hestitate to ask more questions if you think you need more help.

  • $\begingroup$ Thank you, but I don't how to proceed from here. Could you write some steps, please? $\endgroup$
    – Mark
    May 25, 2016 at 16:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.