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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?

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    I think it should say $|\int fdm|\leq|m|(X)||f||$, which is the inequality that guarantees the continuity of the integral operator (linear functional) $f\mapsto \int f dm.$ – 2017-01-07
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    I agree with what you say, but I'm not sure that I can just justify something by saying that it probably means something else as I have found several papers that include the inequality as I wrote it, hence my confusion. Here's an example, perhaps it helps in trying to understand what is meant by the notation: It is taken from Brooks and Lewis (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0338821-5/S0002-9947-1974-0338821-5.pdf). Page 10, includes the following definition: – 2017-01-09
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    "If H is a locally compact Hausdorff space, then an operator $L:C(H,E) \rightarrow F$ (where $E, F$ are locally compact) is said to be dominated provided there is a positive linear functional $P\in C(H)^*$ so that $\|L(f)\| \leq P(\|f\|)$, $f\in C(H,E)$". (Please help!) – 2017-01-09
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    It's a pain when you get stuck by notation, isn't it? I don't know if $||L(f)||\leq P(||f||)$ is another typo or not. Could you give me the reference of the article(s) where your first question shows up? If you have more examples like the previous one please point them out as well. – 2017-01-10
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    Since this is a very old topic I would bet that there is a book or survey about it. I don't know any. Do you? – 2017-01-10
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    The original reference is: https://projecteuclid.org/download/pdf_1/euclid.ijm/1256048836 (look at the second paragraph) – 2017-01-10
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    The reference I mentioned above is https://www.jstor.org/stable/1996826. Thanks for your help! – 2017-01-10

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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.