On a locally compact separable metric space, the finite signed Borel measures with the total variation norm form a real Banach space, because the Riesz representation identifies them isometrically with the dual of the continuous functions with compact support.
In the setting of The Dual of C_c(X): Standing Notation, let be a locally compact metric space that is separable.
Completeness. The space with the total variation norm , a real normed space by Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (vii), is a real Banach space.
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