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The Mathematics Of Happy New Year

My 2012 New Year Wish has some beautiful mathematics: beautifully simple, beautifully abstract, and beautifully mysterious. Want to peek under her skirts?

2: Warm-Up

<$$ \sum_{n=0}^{\infty} \frac{1}{2^n} = %\lim_{n\to\infty} \sum_{k=0}^{n} \frac{1}{2^n} = %\lim_{n\to\infty} \frac{1-\left(\frac{1}{2}\right)^{n+1}}{1-\frac{1}{2}} = %\frac{1}{\frac{1}{2}} = 2 $$>

This is a simple example of a convergent infinite series, that is, an infinite sum of numbers that adds up to a finite number. The notation means the same as

<$$ 1 + \frac{1}{2} +\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots = 2. $$>

You can also imagine the series as a sum of rectangles.

Illustration of the convergent series

0: Functional Analysis

<$$ |\{ (X, ||\cdot||): \text{Banach space & dim } X = \aleph_0\}| = 0 $$>

This dense bunch of symbols expresses one result of functional analysis: there cannot be a Banach space with a countably infinite Hamel (algebraic) basis. (Banach spaces are complete metric spaces, i. e. spaces in which every Cauchy sequence has a limit.) The only possible Banach spaces have either finite dimension (e. g. <$\mathbb{R}^n$>) or uncountable dimension (e. g. <$L^p$>).

The New Year wish contains a concise proof by contradiction of this fact. The existence of a complete space with countably infinite dimension contradicts Baire’s theorem.

1: Complex Analysis

<$$ -e^{i\pi} = 1 $$>

Some trigonometric identities will never cease to fascinate me. Would you expect that both sine and cosine are just parts of the exponential function and all their properties stem from there? Why are the ratios of right triangle’s sides related to the exponential? Mathematics is rich with such surprising connections between seemingly distant subjects. Every such revelation inspires great sense of admiration and rewards you with deeper understanding.

Whether you swear by mathematics or not, allow me to wish you a 2012 full of deep understandings.

January 4, MMXII — Mathematics.