Necessary and Sufficient Condition for Diagonal Operator to be Invertible

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $\sequence {\lambda_n}_{n \mathop \in \N_{> 0} }$ be a bounded sequence in $\mathbb F$.

Let $\ell^2$ be the $2$-sequence space.

Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space on $\ell^2$.

Let $\Lambda \in \map {CL} {\ell^2}$ be the diagonal operator such that:


 * $\forall \mathbf x := \tuple {x_1, x_2, \ldots} \in \ell^2 : \map \Lambda {\mathbf x} = \tuple {\lambda_1 \cdot x_1, \lambda_2 \cdot x_2, \ldots}$

Then $\Lambda$ is invertible in $\map {CL} {\ell^2}$ $\ds \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} } > 0$ where $\inf$ denotes the infimum.