thoughts about the Rιesz Representatiοn Theorem.

i thought of this some months ago, but hadn't bothered to write it until now. so let it be said: never underestimate the likelihood of procrastination. q:

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in calculus we teach students how to compute derivatives and possibly about what it means, geometrically, for a function to be differentiable at a point. but we give only a cursory nod to the notion of continuity; it's usually brushed aside in the standard repetoire, at least.

in such a case, calculus comes first. but any mathematician worth his salt will realise that differentiability is a rare property. in a first course in analysis or topology, continuity is the name of the game; calculus then gets the cursory nod.

so let us ponder continuity of functions.

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in a first course in linear algebra, most examples of linear spaces are copies of euclidean space Rⁿ of varying dimension. the norm is always the standard 2-norm. if one is taught abstract linear spaces, then polynomials of degree n serve the same purpose as Rⁿ; the same norm of yore will do.

but if you have a daring lecturer, then (s)he will give you the example of the class of continuous functions on the closed interval [0,1]; it's a linear space, and i'll call it C^o for short. apply the max-norm, and you have a normed linear space.

in linear algebra, there is also a notion of duality: what linear functionals act on vectors in a continuous manner). in finite dimensions, dual spaces aren't that fascinating and the notion of "weak convergence" is unnecessary, but many strange novelties occur if one moves to infinite dimensions.

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here is what i find to be the real kicker, though. thus far, our discussion about C^o has been topological and linear algebraic in nature. other than a norm, there's not much analysis going on.

now invoke the Rιesz Representatiοn Theorem (or some version of it): the dual space of C^o is precisely the space of measures μ on [0,1], where the action is by integration:

f → ∫_[0,1] f dμ

in other words, applying the notion of linearity and "length" to continuous functions will produce calculus, in the form of measure and integration.

this still astonishes me, to this day. analysis seems to come out of nowhere, and somehow as a natural outcome.

good memory == smart?

EDIT (AS OF TUESDAY @ 9 PM): apparently i don't know euler's formula, either. i suppose it's all the more reason why i should be impressed by those who do.

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today's calc ii class was a review/refresher on how to manipulate taylor series, and i informed my students that certain series are good to remember.

then one student asks about the homework. to start that particular problem, i begin writing down the Taylor series for sin x about x = 0 .. or maybe it was e^x; i forget.

what i do remember is turning around, ready to start an explanation, and there is already a hand up. preparing for the worst, i call on the owner of the raised hand.

"you knew the taylor series of that function, off the top of your head?"

"er, yeah."

"wow."

"well, write it down enough times, and it tends to stick."

then someone else asked if we need to know that series for the exam. i think i caused a minor tumult by saying "yes, and these three others."

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what surprised me was .. well, the surprise. i mean, it's just a formula, and it's a well-known series. it's like knowing that the harmonic series diverges.

i dare say that any mathematics major or grad student worth his or her salt would know the taylor series for e^x off the top of their head.

would you be impressed if you met someone who remembers:

the quadratic formula?
or the pythagorean theorem?
or that sin(π/6) = 1/2?

i might be a little delighted if someone did know euler's formula:

e^iπ - 1 = 0.
e^iπ + 1 = 0.

for instance, the guy from xkcd knows it. great comic, that one.

to be 'educated'

i actually wrote this as a footnote on another blog post, but i think it stands on its own:

to some, being 'cultured' or 'educated' means being well-read in literature, being able to talk philosophy and politik and human nature, being able to appreciate classical music and fine art.

my demands on being 'educated' include those and more. why ignore our techie side? for example, an 'educated' person should be able

1. to appreciate why the harmonic series Σ_n n^-1 diverges,
   
   
2. to appreciate the classical proof of why there are infinitely many prime numbers.
   

that doesn't mean i ask such persons to prove these results. i just ask that they be listeners without glazed eyes. if we should see beauty in symphonies and in baroque art, we should also see beauty in geometry and in number.

less demandingly, i would ask that an 'educated' person be able to build a simple webpage.

it would also be fair to ask that such persons know a little science, but then i risk not being an educated person! q:

post grading blues.

an hour ago (~1:40am) i was still grading calc ii exams. between that and the bike ride home through the rain, i think i've lost a large dose of humanity.

i feel cold and cold-blooded, cynical, and worn out, too. i feel like impassive stone.

all that frustration at not having any time for research has somehow vanished; maybe it lies hidden under too many hours of waking and too many instances of misapplied convergence tests to infinite series. i don't feel the past few nights of fitful sleep and insomnia .. not yet, at least, and i fear they lie in wait for the morning.

argh. tomorrow morning is in a few hours!

my hair is wet and my eyes are wide open; in an hour i will be dead to the world and in a few more hours, jolted awake by necessity and past promises.

i hate having to be responsible. more responsibility means less sleep. \:

time mismanagement.

i should learn to plan more efficiently.

for instance, my friday nights are almost always free. if the week ends well, then i partake in beer with fellow math grads until early evening, and then i go my separate way to be alone for a while.

i never get much work done then, anyways. i should be grading calc ii homeworks then, so that i wouldn't have to grade them now.

you see, i really, really don't want to grade them now.

instead, i would rather be reading from a text by Dupré and Gillette about Banach modules, or lounging in my recliner and reading the latest issue of Bon Appétit magazine.

argh. work comes before .. er, more enjoyable work, and of course, play.

sometimes i really, really hate mathematics.

i'm not sure how it happened, but during my meeting with the advisor i screwed up the same proof two weeks in a row. it's not a new theorem, and not even the proof of a theorem, but a single line in a proof.

one. single. line.

i hate being inept, i hate wasting the advisor's time, and for a little while today, i hated mathematics.

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after that debacle of a meeting and two seminars later, i sat at my office desk and thought. in other words, i stared at the chalkboard and it stared right back.

then i decided to make a cup of coffee, and as i was pouring the water into the back chamber of giulia [1] the idea suddenly hit me. by the time the coffee was ready, i came up with a correct proof ..

.. well, it looks correct, but so did the last two 'proofs' of mine. but looking at these steps, it's obvious. f*cking obvious.

so i sipped the coffee angrily, hardly enjoying it at all.

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[1] yes, the coffee maker is called giulia, and no, i didn't name it; my officemate kevin tu¢ker did.

when we were trying to come up with names, i suggested we could name it after a beautiful woman and immediately, he thought of giulia (pronounced: julia, but trochaically).

a grad population, over the years.

a first-year grad student, my desk was in 1061 east hall with a dozen others. just earlier i remembered those former officemates of mine:

• i'm still officemates with ryan, after all these years. mike, whose desk was across from me, is now a little down the hall from my current office. marie and jared had corner desks, that year; he's now down the hall and she's now on the fifth floor.
  
  
• john is still a um student, but his advisor is at yale, so he is there for the year (or longer).
  
  
• i only see kyung yong when i am walking to the library or moving up- and downstairs. young kwon is still somewhere in ann arbor; few ever see him.
  
  
• eliana transferred to chicago, to be closer to her husband.
  
  
• alison returned to her home state of florida.
  
  
• dave decided to be an ecologist/biologist. he's now on the other end of the diag, and northwest of east hall.
  
  
• ken is no longer with us.
  
  
• ray, yuan, and another chinese student finished their actuarial programs, and are now working people.
  

much has happened in three years. the world turns, the years pass, and we grow older and have more stories to tell. the new kids still look fresh-faced, even the second-year students.

they haven't seen their peers leave .. yet.

in which i read the funnypages.

last night i (re)discovered a webcomic called xkcd. i don't know what the title means, but every so often they have these hilarious mathematical/nerd comics.

for example, [1]

full-sized image
link to this comic

and these also crack me up, even if physics are involved:

link to this comic


(can't find the link for this one, but it's on the website)

[1] this artwork is NOT mine, but the work of one Randall Munroe. You can read about him here.