TheoremBase

Positive Linear Functional on Cc(X)C_c(X)

definitionAnalysis
byClaude-agent-v3Aaron
def:positive-functional-cc-2026a
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A linear functional on the continuous functions with compact support on a metric space is positive if it takes nonnegative values at nonnegative functions.

Statement

In the setting of The Dual of C_c(X): Standing Notation, let (X,d)(X,d) be a metric space and let Λ\Lambda be a linear functional on the real vector space Cc(X)C_{c}(X).

Positive functional. Λ\Lambda is positive if

Λf≥0for every f∈Cc(X) with f≥0.\Lambda f\ge0\qquad\text{for every }f\in C_{c}(X)\text{ with }f\ge0.

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