Notation regarding vector measures

Question

I have a simple question, regarding notation that I just haven´t been able to solve even though I have tried looking at different sources. The problem is this: Let X be a compact Hausdorff space and E a Banach space, then $C(X,E)^*$=$M(X,E^*)$ where $C(X,E)$ are the continuous functions from X to E and $M(X,E^*)$ are regular Borel measures with finite variation. So if we take one of these measures m, and |m| denotes its variation I have come across the following: For any f:X$\rightarrow$ E that is continuous, $|m(f)|\leq |m|(\|f\|)$ and it is this last inequality I don't understand... is this notation common? Can someone please explain?

Answer 1

Mystery solved! In the inequalities $|m(f)|\leq|m|(||f||)$ and $||L(f)||\leq P(||f||)$ the symbol $||f||$ represents the function $x\mapsto ||f(x)||$.

Evidence 1: In the paper GROTHENDIECK SPACES (KHURANA, p.79) the inequality $|\mu(f)|\leq|\mu|(||f||)$ is stated for $\mu\in (C(X,E))'=M(X,E')$ making clear reference to the identification between the functional $f\mapsto \int fd\mu$ and the measure $\mu$ via the representation theorem. Therefore the inequality $|\mu(f)|\leq|\mu|(||f||)$ means $|\int f d\mu|\leq\int ||f||d|\mu|$ which appears in the proof of Theorem 2.8 of LINEAR OPERATORS AND VECTOR MEASURES (BROOKS AND LEWIS, p.148) and in BOUNDED CONTINUOUS VECTOR-VALUED FUNCTIONS ON A LOCALLY COMPACT SPACE (WELLS, P.121).

Evidence 2: The paper INTEGRAL REPRESENTATION THEOREMS IN TOPOLOGICAL VECTOR SPACES (SHUCHAT, P.391), below the inequality (20), states that $||f||$ represents the function $x\mapsto ||f(x)||_E$.

Evidence 3: The paper STRICT TOPOLOGY AND PERFECT MEASURES (KHURANA and VIELMA, p.2) defines explicitly the symbol $||f||$ as the function $x\mapsto ||f(x)||$.

Thus the inequality $||L(f)||\leq P(||f||)$ makes perfect sense for $f\in C_0(X,E)$ and $P\in C_0(X)'$ a positive linear functional, because $||f||$ (under the mentioned definition) belongs to $C_0(X)$.

PS: S.S. Khurana was the PhD adviser of my MSc adviser José Aguayo Garrido. I worked with strict typologies in my master thesis. A lot of fun.