Hello,

I just can't see why the following operator is well-defined: Let $\displaystyle x \in \ell^{2}(\mathbb{N}), a \in \ell^{1}(\mathbb{N}) $. Define the operator $\displaystyle A$ so that $\displaystyle Ax = (\sum_{k=1}^{\infty}a_{k+n-1}x_{k})_{n \in \mathbb{N}}$. Also, assume that $\displaystyle a_n$ is monotonically decreasing. Can anybody show me why $\displaystyle A: \ell^{2}(\mathbb{N}) \rightarrow \ell^{2}(\mathbb{N})$ is supposed to be well-defined?

Thanks for your help,

Best regards,

Stiwan