... "i sat in abject horror, my mathematical blood ran cold" ...

I will show you something different from either
        Your shadow at morning striding behind you
                Or your shadow at evening rising to meet you;
                        I will show you fear in a handful of dust. [0]


you know, i had always been dismissive of those constructivists who, among other things, refuse to accept the axiom of choice as part of their proof-writing toolbox: a bunch of mealy-mouthed naysayers and contrarians, i thought!

it had always seemed to me a handy, albeit strange, tool .. but it gets the job done, right?

one of its consequences, the hahη-baηach separation theorem, is incredibly useful .. if not highly magical and never leading to any concrete example. if i can't build something by hand, then usually i use hb.

similarly, for me baηach-alaοglu is like crack: i think i have some mental addiction to weak-star convergent subsequences .. or, if the situation calls for it, nεts. [1]

the last few days have been conceptually difficult .. to the point where i thought i stumbled onto either a paradox or a counterexample to one of my own results.

i felt like my imagination was being stretched to its (very limited) capacity and that i was teetering over the edge of conventional sanity. this afternoon i attained some kind of resolution, though. at the same time,

i sat in abject horror,
my mathematical blood ran cold,
and i was ready to throw down some printed pages to the ground,
step on them repeatedly,
and immediately afterwards, run away, screaming.

fortunately (for my officemate, anyway) and as creepy as this feeling was, i restrained myself and sat calmly at my desk, trying to look at the bright side:

well, at least the theorem's not wrong.

---

for the record, i've been thinking (too much and too often) about the dual of the βanach space of functions $L^\infty(\mathbb{R}^n)$, which consists of bounded, fιnitely-additive sιgned measures on $\mathbb{R}^n$ that vanish on sets of leþesgue measure zero.

if you have never been curious about these objects, then don't start now. seriously.

although they have a not-unnatural role in functional analysis, my obsession with them has gotten to the point
where i think i have become a generally worse person and perhaps less human.

i learned of the following results from an old paper of hewιtt and yοsida, called finitely additive measures from 1952. in that sense, it reads like a gothic novel .. all quiet and calm at first, and then the monsters come.

theorem 3.3: fubini's theorem fails for finitely additive measures in $[L^\infty(\mathbb{R})]^*$.

theorem 3.4: there exists a nonzero finitely-additive measure $\zeta$ on $\mathbb{R}$ so that $$ \int_{-\infty}^\infty c(t) \, d\zeta(t) \;=\; 0 $$ for all bounded continuous functions $c$ ... and any such measure $\zeta$ must be purely finitely additive.

roughly speaking .. by "purely finitely additive" here, they mean that the object cannot have any nonzero part that behaves like a usual (countably additive) measure. in other words, it's a distinctly exotic object.

theorem 3.6. for any real number $a$, there exists $\zeta_a$ in $[L^\infty(\mathbb{R})]^*$ so that $$ \int_{-\infty}^\infty x(t+u) \, d\zeta_a(u) \;=\; x(t+a) $$ for all essentially bounded functions $x \in L^\infty(\mathbb{R})$ and a.e. $t \in \mathbb{R}$ [2].

keep in mind that $\zeta_a$ is not a point-mass at $a$. in particular, it vanishes on all βorel sets of leþesgue measure zero! (this, by the way, was something close to the paradox i had in mind, believing that it was impossible ..)

thinking about it, the results aren't that much more surprising than the banach-tarsκi paradοx. then again, it's been an obsessive week or two.

i think i need a vacation.




[0] re-reading this line by eliot, notions like "cantοr dust" and singular measures come to mind.
[1] not everything in life is metrisable, you know. for some reason, my work has taken me to these exotic locales, lately.
[2] the original statement was over-simplified. thanks to L for pointing this out.

initiating self-destruct countdown: ... 6, 5, 4 ...

.. a week or two ago, i had an idea for a new theorem;
yesterday i was about to put the polish on the proof ..

.. and this morning, i almost constructed a counterexample for it,
with an emphasis on the word "almost" ..!

[sighs]
i guess it's another session at the library today ..

the more i learn about duality in Banach spaces, the subtler it seems to become.

in medias res: a three-star rating .. (UPDATED).

// initially posted: 2013-03-07 @ 13:34EEST

odd.

today i took the double dual $(V^*)^{**}$ of a certain dual Banach space $V^*$.
it was rather confusing, but seemed like a good idea at the time ..

// updated: 2013-03-08 @ 13:30EEST

i think i've either jumped off the deep end or gotten nostalgic for topics i learned during my student days. in the last two weeks i've gone to the library four times to read the books on functiοnal analysιs by rudin and by dunford-&-schwartz.

more and more i am impressed by the latter volume, but that's more an artifact of my current mathematical necessities; lately, you see, i have become wont of more gory details regarding some seeming familiar Banach spaces. this sounds like pure folly ..

.. but i want to understand better the space $L^\infty(\mathbb{R}^n)$ and its Banach dual! [1]

there is already a handy characterisation, but it uses words like "finitely additive" and "set function" .. which, admittedly, worry me. the awesome thing about D&S is that they have an entire chapter written in terms of these things. apparently, most of the usual measure and integration theory runs analogously through.

more to the point, i don't have to re-prove everything myself; these guys are lifesavers!

on a related note, these two passages from D&S caught my eye [2].

1. Theorem III.5.13 & 14 (Алексaндров). [If $\mu$ is] a bounded, regular, complex-valued, [finitely-]additive set function defined on a field $\Sigma$ of subsets of a compact topological space $S$, [t]hen $\mu$ is countably additive ... [Moreover] there is a unique regular, countably additive extension to the $\sigma$-field determined by $\Sigma$.
   
   i'm probably too naive, but it's hard for me to appreciate the difference between finitely additive measures and the usual (countably additive) ones. in light of this theorem, though, there are two points that come to mind:
   
   (A) once the set function is assumed regular --- i.e. that it has "good limits" for sets that fit the topology of the space --- then on compacta, there is no difference between finitely- and countably- additive. on the other hand, set functions that are strictly outside of $L^1(\mathbb{R}^n)$ must therefore lack these nice limit-properties, which means that they will be hard to work with!
   
   (B) this seems to be one of the rare criteria for checking whether a bounded linear functional of $L^\infty(\mathbb{R}^n)$ is actually a Lebesgue integrable function .. or, more precisely, part of a criterion. the theorem guarantees that we can work with measures as usual, so the Radon-Nikodym theorem would apply in this setting.


2. some parts of the book provide suggestions and instructions for how to proceed. for example, on pg. 122, III.3.6 it reads:
   
   The reader will more easily perceive the significance of the somewhat complicated conditions (ii) and (iii) in the following [Theorem 6], if [s]he reads the statement and proof of Theorem 7 after the statement of Theorem 6 but before its proof.
   
   wait .. so why didn't they just switch the order of theorems 6 and 7? (-:

// updated: 2013-03-08 @ 17:30EEST

the further i read on, the more uneasy i feel. some of these constructions are just .. creepy: i can't think of another way to put it.

one result is outright unnerving:

Theorem IV.6.18-19: If $S$ is a compact Hausdorff space, then there exists a(nother) totally disconnected, compact Hausdorff space $S_1$ and an embedding $i: S \hookrightarrow S_1$ so that $i(S)$ is dense in $S_1$ and that induces an isometric isomorphism $$ i_*: L^\infty(S,\mu) \,\to\, C(S_1). $$

to get a sense of what i mean, one can take as nice of a space $S$ as possible and extend it to some $S_1$ so that $S$ is dense, yet incredibly scattered within $S_1$ .. to the extent that characteristic functions $\chi_A$ of subsets $A$ in $S$ extend to continuous functions in $S$.

the real kicker is that the extension is also a characteristic function of some subset in $S_1$ as well (IV.9.10). roughly speaking, then, the continuous extension is not done through interpolating between the values $0$ and $1$, but by sufficiently separating the preimage sets in $S_1$ between these values .. and none of the subsets $A$ are chosen in advance!

in short, $S_1$ must be a weird, messed-up space.

..
..
.. i think i'm going to stop reading for today.

[1] .. and in case you're keeping count, no: $[L^1_\mu(\mathbb{R}^n)]^{**} \cong [L^\infty_\mu(\mathbb{R}^n)]^*$ is only two stars. (i might get to the third one in a later update.)

[2] the square brackets []'s indicate my reformatting.