Riesz Basis Of Hilbert Space

Question

We know that $\{x_k\}$ is a Riesz basis for $~H~$ if and only if there are constants $~0 ≤ A ≤ B~$ such that for all finite sequences $${α_k},A||x||^2\leq\sum_{k}\langle x,x_k \rangle^2\leq B||x||^2.$$ My question: " Is every orthonormal basis a Riesz basis? "

Answer 1

(You have not stated the definition of a Riesz basis correctly). For an orthonormal basis $(x_k)$ we have $\|\sum a_k x_k\| ^{2}=\sum |a_n|^{2}$. So $(x_k)$ is a Riesz basis with $A=B=1$.

Answer 2

Yes, and your can verify that $A=B=1$ for an orthonormal basis.