Measure, Integration & Real Analysis · Problem 6D.3

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · linear-functionals-null-space-inclusion

Problem

Suppose φ\varphi and ψ\psi are linear functionals on the same vector space. Prove that null⁡φ⊆null⁡ψ\operatorname{null} \varphi \subseteq \operatorname{null} \psi if and only if there exists α∈F\alpha \in \mathbf{F} such that ψ=αφ\psi = \alpha \varphi.

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