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.