Name: Real and complex analysis
ISBN: 9780070542341

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graycatJul 2025

"Functions Are Vectors"?

Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f

f: A --> N

with usual notation.

Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.

So, is X a vector space and, thus, each f in X a vector?

No, since -f is not in X. Neither is (1/2)f.

Some references (with TeX markup):

Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}

linear algebra treated as functional analysis.

Walter Rudin, {\it Real and Complex Analysis\/}

with Lebesgue integration and, then, Banach and Hilbert vector spaces.

Walter Rudin, {\it Functional Analysis\/}

with Fourier theory.

Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}

with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.

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graycatApr 2024

Consider ordinary 3 dimensional space: Go to a corner of the room, where the floor and two walls meet, and see three lines, each line the intersection of the two walls or a wall and the floor. In usual building architecture, these three lines are mutually perpendicular, i.e., orthogonal.

Regard each of these three lines as an axis of a coordinate system. In particular, pick a point on each of the three axes to give us points with coordinates:

(1,0,0) -- the X axis

(0,1,0) -- the Y axis

(0,0,1) -- the Z axis

Soooo, now we have a standard coordinate system for ordinary 3 dimensional space.

Now consider a point P in the room (in my case, the end of the pony tail of that sweet, gorgeous blond I keep looking at!) with coordinates

(x,y,z)

for some numbers x, y, z.

For the X coordinate of point P we make use of the X axis and our point P: We take the dot (inner) product of

(1,0,0) and (x,y,z)

and get, presto, bingo, wonder of wonders, no doubt, x.

Similarly for y and z.

Or, for each axis, our inner product projects point P on that axis. And from those projections we can reconstruct the coordinates of point P. Beginning to sound like Fourier theory????

This stuff of projections onto orthogonal and then reconstructions from that generalizes.

In Fourier theory, those sine waves are mutually perpendicular -- project one on another and get 0. That summation is the inner product for that projection.

And if not a summation, then an integration -- same stuff, use an inner product to do projections onto mutually perpendicular axes.

For the finite stuff, and just summation signs, it's fairly elementary.

For the integral stuff, with the Riemann integral of calculus, can see Baby Rudin, Principles of Mathematical Analysis and Fourier series, over finite lengths of time. For the Fourier integral, over the whole real axis, see the Lebesgue integral (partition on the Y axis instead of the X axis and get nicer properties) in Rudin's Real and Complex Analysis. For a version that uses Schwartz distributions, there Rudin's Functional Analysis.

Net, it's all about inner products and projections as we saw in the corner of a room.

Soooo, for some of the generalization, are working in an inner product space. A nice introduction is, of course, Halmos, Finite Dimensional Vector Spaces: Soooo, that's ordinary linear algebra, finite dimensional, but done with techniques that mostly generalize to infinite dimensions, e.g., as in Fourier series, integrals, distributions.

Rudin's Real and Complex Analysis also does Fubini's theorem that helps with doing Fourier theory on surfaces, e.g., the ocean.

History: Halmos had gotten his Ph.D. under Doob, of Stochastic Processes (e.g., the stock market, weather, medical EGK, ...), and got a position under John von Neumann at the Institute for Advanced Study in Princeton; so, apparently Halmos wrote his finite dimensional book borrowing from von Neumann's work on the infinite dimensional case, e.g., as in von Neumann's Quantum Mechanics.

Right, a complete inner product space is a Hilbert space -- the complete is essentially as also in the real numbers, sequences that appear to converge in the distance (from the inner product) used always do have something to converge to.

Sooo, Hilbert space has a lot, asks a lot, making the examples restricted -- Rudin has a nice theorem that shows that really there's only one Hilbert space!

Halmos is my favorite author. Rudin is a close second.

Education Technique: While starting my career in computing, I studied both Halmos and Rudin carefully, independently. In linear algebra, also E. Nering. At the time, did a lot with the FFT (fast Fourier transform, mostly of J. Tukey), and, thus, at one point got "sole source" on some work for the US Navy. Then in grad school, the backgrounds in Halmos, Nering, Rudin, and the computing got me the best in the class on the three Ph.D. qualifying exams -- linear algebra, analysis, computing.

As I recall, Nering was a student of E. Artin, at Princeton. Nering's book also considers the case of finite fields, that is, not only the rational, real, complex numbers. That helped in a course in error correcting codes.

Why is the FFT "fast"? Well, if all you want is some one Fourier coefficient, it's not "fast". If you want all the coefficients, then just write a Do-loop and find them all one at a time -- easy enough to do. Get the discrete Fourier transform. But if go for all the coefficients this way very often on fairly long streams of data, then can slow down even a fast computer. BUT: If the number of points in time is a power of 2, commonly 1024 = 2^10, and look carefully at elementary trigonometry, can see how to do a little on all the computing, then a little more on all of it, ..., for ~10 such passes, and, presto, bingo, be really a LOT faster, essentially n log(n) instead of n^2.

There's more at

https://www.youtube.com/watch?v=nmgFG7PUHfo

with the role of Richard Garwin working to do Fourier analysis on seismic data as part of detection of underground nuclear tests.

Much of the faculty was really, REALLY torqued that (1) I didn't fail the exams and (2) led the class. REALLY torqued!

Halmos, Rudin -- can be good stuff.

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graycatAug 2023

Sorry! I LOVE measure theory and find it much better and especially EASIER than traditional Riemann integration!!!

In simplest terms, do the partitioning on the Y-axis (the range of the function being integrated) instead of the X-axis (the domain). Right away that makes good sense because we KNOW the range is just the real numbers, and partitioning on the range frees up the domain to be quite general, with a sigma algebra, e.g., where do the sampling as in probability theory.

In one step further, get to f'get about the domain of the function being a compact set and the function being continuous and uniformly continuous -- just get to set all that aside. Also get a straight forward theory of the domain being, say, the whole real line without any extra work, without the Riemann approach where have to take a limit as the limits of the integral go to infinity. And, again compared with Riemann, the measure theory limit that defines the integral is simpler and makes more sense.

For an extra goodie, get a nice, unusually general proof of Leibniz's rule, that is, differentiation under the integral sign!

W. Rudin's Real and Complex Analysis has a really nice, succinct, clean (precisely reasoned) treatment early in that book.

Maybe what I have explained here can provide enough of an overview and some intuition to make the Rudin material quite easy to read.

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ecesenaJul 2023

The preface of Real & Complex Analysis by W. Rudin.

It shows that exp is periodic, the first time I read it kind of blew my mind.

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graycatFeb 2023

> The unreasonable effectiveness of conditional probabilities

Yup.

There is a more general result that is more amazing with still more "unreasonable effectiveness":

For events A and B, we can consider the conditional probability of A given B:

P(A|B) = P(A and B)/P(B)

Then we can generalize P(A|B):

Suppose we do an experiment and make two observations, X and Y. We regard each of X and Y as a random variable that takes on real number values.

We assume that X and Y have averages, that is, expectations, and write those as E[X] for the expectation of X and E[Y] for the expectation of Y.

We want to use X to approximate Y.

So, we want a real valued function f of one real variable x and use f(X) to approximate Y.

For the most accurate approximation, we want to minimize the expectation

E[(Y - f(X))^2]

Claim: For f(X) we want

f(X) = E[Y|X]

where E[Y}X} is the conditional expectation of Y given X and a generalization of P(A|B).

Proof:

We start by using one of the properties of conditional expectation and then continue with just simple algebra:

E[(Y - f(X))^2]

= E[ E[(Y - f(X))^2|X] ]

= E[ E[Y^2 - 2Yf(X) + f(X)^2|X] ]

= E[ E[Y^2|X] - 2f(X)E[Y|X] + f(X)^2 ]

= E[ E[Y^2|X] + E[Y|X]^2 - 2f(X)E[Y|X]

+ f(X)^2 - E[Y|X]^2 ]

= E[ E[Y^2|X] + ( E[Y|X] - f(X) )^2 - E[Y|X]^2 ]

which is minimized with

f(X) = E[Y|X]

Done.

So, the "unreasonable effectiveness" of conditional expectation that it provides the best non-linear least squares approximation.

In terms of statistics E[Y|X] is the minimum variance estimator of Y and is also unbiased.

Yes, P(A|B), conditional probability, can be seen as a special case of E[Y|X], conditional expectation.

The detailed definition of E[Y|X] is based on the Radon-Nikodym theorem. W. Rudin's Real and Complex Analysis gives the novel von Neumann proof. Then in Neveu, Mathematical Foundations of the Calculus of Probability is elegant coverage of E[Y|X].

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jordibcJan 2023

For sure I got significantly(?) better with classics like Spivak, Apostol, Rudin.

"Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me.

Other than more "classic math" books, I also wanted to mention two outliers that I found eye-opening and generally awesome:

* Street-Fighting Mathematics, by Mahajan (http://streetfightingmath.com/). Intuitive, useful and fun.

* Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.

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