For a given Hilbert space find a tight frame with bound A

Question

For a given Hilbert space and $A>0$ find a tight frame with bound A.

I know that an ortho-basis is a tight frame with $A=1$. Can I extend this to any $A>0$ by just scaling the ortho-basis?

Answer 1

Yes, you can. If $\{e_\alpha:\alpha\in I\}$ is an orthonormal basis, then $\{\lambda e_\alpha:\alpha\in I\}$ is a tight frame with constant $A=\lambda^2$: $$ \sum_\alpha |\langle v,\lambda e_\alpha \rangle|^2 = \lambda^2\sum_\alpha |\langle v, e_\alpha \rangle|^2 = \lambda^2\|v\|^2 $$