- Two Basic Theorems
- Adjoints
- Invertible Operators
- Non-Emptiness of the Spectrum
- Spectral Radius
- Self-Adjoint Operators
- Positive Operators
- Operators on Tensor Products

This book began as lecture notes taken by William Paschke's functional analysis students at the University of Kansas in 2002. It was his vision (as presented in his original introduction below) that they should eventually become more. Ten years later I asked Dr. Paschke to go through the notes again. He did so with an eye toward preparing a textbook from them, adding in more detail and additional topics. The result is a unique view of the subject with examples hard to find anywhere else in an introductory book. All of the basics and needed information are given, but many of the details are left to the reader to verify, making this also a good book to learn on your own.

These notes are also intended as an example of the benefits of XML authoring of mathematics. I was inspired in this by Rob Beezer's PreTeXt project, and this book attempts to go one step further by encoding not only the document structure but the mathematics as well in XML. You can hover over many symbols and click on others for knowls of definitions to appear. Comments and corrections are welcome (l3c@orthogonalpublishing.com).

One important aspect still missing in this early version is the sense of Bill's humor and style (that made his teaching so enjoyable) that the class embedded in the original notes. Edition 1.0, when we get there, will reinstate the marginal comments.

Lon Mitchell, 2023

I’ve been browsing through the presentation editions of the 961 notes.

My lawyer agrees with me that the marginal comments are completely outrageous and over the top, but believes there’s nothing actionable here, since I apparently actually said these things in front of witnesses. Seriously – I mean it! – we should pause to consider the fate of this document as intellectual property. First of all, it ought to belong to you guys rather than to me. Having thus dealt myself out of the movie rights, I must add that I wouldn’t mind seeing it gain somewhat wider exposure. The traditional print channel offers only the pleasures of martyrdom: rendering of the text into standard mathematical prose; pumping in a few gallons of superfluous scholarship to show we’re not just making it all up; enduring the bad manners of editors and reviewers; waiting forever and ever; and, finally, under the best of circumstances, losing sleep over our market share versus Young’s book. On the web, however, would be perfect. As is. I’ve looked it over plenty closely enough to vouch for its essential soundness. In all modesty, I think it’d be as good a source as any for someone looking for an efficient presentation of the basics of Hilbert space and operators thereon, with a tilt toward compact operators.

So … HOW on the web? I know nothing of such matters technically, but in editorial terms, it might work pretty well as a contribution to a departmental electronic lecture notes series, on the same footing as our existing preprint series. Everything speaks in favor of the creation of such a series, and nothing against – except inevitable accusations of egomania if I push too hard for it myself. It’s something to think about, anyway.

Whatever happens, my original expectations for the notes have been overfulfilled by a couple orders of magnitude. All I really had in mind when we started the semester was the photocopying of handwritten notes, as in my own graduate student days during the Harding administration (except then we inscribed our product on stencils and mimeographed it, which old-timers marvelled at as a vast improvement over cuneiform).

Great job! Over to you all. AND LET US NOT FORGET, the real star of the show is good ol’ Hilbert Space.

Bill Paschke, 2003

Here is a motivating example from Ordinary Differential Equations: For a continuous function $\( φ : \mathopen{}\left[0, 1\right]\mathclose{} \to \mathopen{}\left[0, \infty\right)\mathclose{} \)$ and a complex number $\( λ \)$, consider the initial value problem $\( f'' \mathopen{}\left( r\right)\mathclose{}+λf\mathopen{}\left( r\right)\mathclose{}φ\mathopen{}\left( r\right)\mathclose{}= 0 \)$; $\( f\mathopen{}\left( 0\right)\mathclose{}= 0 \)$, $\( f' \mathopen{}\left( 0\right)\mathclose{}= 1 \)$. Let $\( {f}_{λ} \)$ be the solution of the initial value problem corresponding to $\( λ \)$ (see Figure A for an example). Define $\( Φ : \mathbb{C} \to \mathbb{C} \)$ by $\( Φ\mathopen{}\left( λ\right)\mathclose{}= {f}_{λ}\mathopen{}\left( 1\right)\mathclose{} \)$.

Now consider nonnegative $\( λ \)$. When $\( λ= 0 \)$, we have $\( {f}_{0}\mathopen{}\left( r\right)\mathclose{}= r \)$ and thus $\( Φ\mathopen{}\left( 0\right)\mathclose{}= 1 \)$. As depicted in Figure B, as $\( λ \)$ increases, we find certain values force $\( {f}_{λ}\mathopen{}\left( 1\right)\mathclose{}= 0 \)$ .

How we will eventually understand this: using the Hilbert space $\( H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}, φ\mathopen{}\left( t\right)\mathclose{} \, \mathrm{d}t\right)\mathclose{} \)$ we define an operator $\( K : H \to H \)$ by $\[ \mathopen{}\left(K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( s\right)\mathclose{}= {-} \int _{0}^{s}{} ξ\mathopen{}\left( t\right)\mathclose{}\mathopen{}\left(s-t\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t +x\int _{0}^{1}{} ξ\mathopen{}\left( t\right)\mathclose{}\mathopen{}\left(1-t\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t \text{.} \]$ The operator $\( K \)$ satisfies $\( \mathopen{}\left(K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}'' +ξφ= 0 \)$ and is a positive compact operator on $\( H \)$ with eigenvalues $\( \frac{1}{{λ}_{1}} \)$, $\( \frac{1}{{λ}_{2}} \)$, …. In fact, $\( K \)$ is trace-class (meaning the sum of the eigenvalues is finite) with trace $\[ \sum_{j=1}^{\infty}{}\frac{1}{{λ}_{j}} \text{.} \]$