If each of E and F is a real Banach space,
H a compact Hausdorff space, C(H, E)
the Banach space (sup norm ∥·∥∞) of
continuous E-valued functions defined on H,
L: C(H, E)→F a
continuous linear transformation (=operator) with representing
measure m, [sum ] the σ-algebra of Borel subsets of
H and m˜(A) the semivariation of m
on A∈[sum ], then m maps [sum ] into
B(E, F**), the Banach space of all operators
from E
into F** (= the bidual of F),
∥L∥=m˜(H) and
L(f)=∫ fdm. The reader may consult
[9] or [6] for a detailed
discussion of the Riesz representation theorem in this setting.