Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-integral $$\int_X f d \mu$$ exists.
Let $p$ be a continuous seminorm on $E$. Is it true that the inequality $$p\left(\int_X f d\mu\right)\le \int_X (p\circ f) d\mu$$ holds?
Context: This seems to be implicitly used in the proof of lemma 2.4 chapter VI "Left Hilbert Algebras" in Takesaki's second book "Theory of operator algebras".
This is almost correct. Note that $p \circ f$ might not be measurable, which is why the right-hand side has to be replaced by the lower Lebesgue integral. The result then follows from the Hahn-Banach theorem, see Wikipedia for the argument and the definition of the lower Lebesgue integral.