## Real and Complex Analysis, by C. Apelian and S. Surace, now publishedDecember 19, 2009

Posted by Akhil Mathew in analysis, General, math education.
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(This material is reposted from here.)

The book Real and Complex Analysis, by Christopher Apelian and Steve Surace, was recently released.

It’s mainly for an introductory upper-level undergraduate course in real and complex analysis, especially at small liberal-arts colleges.  In this post, I’ll describe this book and how I was involved in its production.

At the start of my freshman year, my analysis teacher, Professor Surace, asked me to check over the drafts of a book he and his colleague (and my former teacher) Prof. Apelian were working on. It was the textbook for the course. At the time, if I remember correctly, there were six chapters: on the real and complex spaces, basic topology, limits, continuity, convergence of functions, and derivatives. The complex analysis part of the book was in its infancy (e.g., there was only a rudimentary outline of one chapter, which had been written some time back and was typeset in Word—they wrote it well before before they had switched to LaTeX). (more…)

## The Hahn-Banach theorem and two applicationsNovember 28, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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I have been finishing my MaBloWriMo series on differential geometry with a proof of the Myers comparison theorem, which right now has only an outline, but will rely on the second variation formula for the energy integral.  After that, it looks like I’ll be posting somewhat more randomly.   Here I will try something different.

The Hahn-Banach theorem is a basic result in functional analysis, which simply states that one can extend a linear function from a subspace while preserving certain bounds, but whose applications are quite manifold.

Edit (12/5): This material doesn’t look so great on WordPress.  So, here’s the PDF version.  Note that the figure is omitted in the file.

The Hahn-Banach theorem

Theorem 1 (Hahn-Banach) Let ${X}$ be a vector space, ${g: X \rightarrow \mathbb{R}_{\geq 0}}$ a positive homogeneous (i.e. ${g(tx) = tg(x), t >0}$) and sublinear (i.e. ${g(x+y) \leq g(x) + g(y)}$) function.

Suppose ${Y}$ is a subspace and ${\lambda: Y \rightarrow \mathbb{R}}$ is a linear function with ${\lambda(y) \leq g(y)}$ for all ${y \in Y}$.

Then there is an extension of ${\lambda}$ to a functional ${\tilde{\lambda}: X \rightarrow \mathbb{R}}$ with

$\displaystyle \tilde{\lambda}(x) \leq g(x), \ x \in X.$

I’ll omit the proof; I want to discuss why it is so interesting.  One of its applications lies in questions of the form “are elements of this form dense in the space”? The reason is that if ${X}$ is a normed linear space and ${Y}$ a closed subspace, the quotient vector space ${X/Y}$ is a norm with the norm ${|x+Y| := \inf_{y\in Y} |x-y|.}$ (The closedness condition is necessary because otherwise there might be nonzero elements of ${X/Y}$ with zero norm.) (more…)

## A theorem of Mazur-Ulam on isometric maps of vector spacesNovember 22, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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I first posted this entry at Climbing Mount Bourbaki, where I have continued the MaBloWriMo series into topics in Riemannian geometry such as the Cartan-Hadamard theorem.  This particular material came up as part of the proof that distance-preserving maps between Riemannian manifolds are actually isometries.  However, the style of the entry seemed appropriate for this blog, so I’m placing it here as well.

The result in question is:

Theorem 1 (Mazur-Ulam) An isometry ${M: X \rightarrow X'}$ of a normed linear space ${X}$ onto another normed linear space ${X'}$ with ${M(0)=0}$ is linear. (more…)

## Riemann integration in abstract spacesSeptember 30, 2009

Posted by Akhil Mathew in analysis.
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