 The above series converges in "L"("X"), which implies the existence of ("z"2−"T")−1, if

Therefore, the resolvent set ρ("T") is open and the power series expression on an open disk centered at "z"1 ∈ ρ("T") shows the resolvent map is analytic on ρ("T").Another expression for ("z"−"T")−1 will also be useful.The formal expression

leads one to consider

This series, the "Neumann series", converges to ("z"−"T")−1 if

From the last two properties of the resolvent we can deduce that the spectrum σ("T") of a bounded operator "T" is a compact subset of C. Therefore, for any open set "D" such that σ("T") ⊂ "D", there exists a positively oriented and smooth system of Jordan curves Γ = {γ1, ..., γ"m"} such that σ("T") is in the inside of "Γ" and the complement of "D" is contained in the outside of Γ.