The open set "G" could vary with "f" and need not be connected. The integral is defined as the limit of the Riemann sums, as in the scalar case. Although the integral makes sense for continuous "f", we restrict to holomorphic functions to apply the machinery from classical function theory (e.g., the Cauchy integral formula). The assumption that "σ"("T") lie in the inside of Γ ensures "f"("T") is well defined; it does not depend on the choice of Γ. The functional calculus is the mapping Φ from Hol("T") to "L"("X") given by