Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

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Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

Forward

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

Introduction

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

Warm-Up

Here is a motivating example from Ordinary Differential Equations: For a continuous function φcontinuous function :maps [interval0zero, 1one]interval →to [interval0zero, ∞infinity)interval \( φ : \mathopen{}\left[0, 1\right]\mathclose{} \to \mathopen{}\left[0, \infty\right)\mathclose{} \) and a complex number λcomplex number\( λ \), consider the initial value problem ffunction″second derivative⁡(rreal number)+plusλcomplex number⁢timesffunction⁡(rreal number)⁢timesφcontinuous function⁡(rreal number)=equals0zero \( f'' \mathopen{}\left( r\right)\mathclose{}+λf\mathopen{}\left( r\right)\mathclose{}φ\mathopen{}\left( r\right)\mathclose{}= 0 \); ffunction⁡(0zero)=equals0zero \( f\mathopen{}\left( 0\right)\mathclose{}= 0 \), ffunction′derivative⁡(0zero)=equals1one \( f' \mathopen{}\left( 0\right)\mathclose{}= 1 \). Let ffunctionλcomplex number\( {f}_{λ} \) be the solution of the initial value problem corresponding to λcomplex number\( λ \) (see Figure A for an example). Define Φfunction:mapsCcomplex numbers→toCcomplex numbers \( Φ : \mathbb{C} \to \mathbb{C} \) by Φfunction⁡(λcomplex number)=equalsffunctionλcomplex number⁡(1one) \( Φ\mathopen{}\left( λ\right)\mathclose{}= {f}_{λ}\mathopen{}\left( 1\right)\mathclose{} \).

Now consider nonnegative λnonnegative number\( λ \). When λnonnegative number=equals0zero\( λ= 0 \), we have ffunction0zero⁡(rreal number)=equalsrreal number \( {f}_{0}\mathopen{}\left( r\right)\mathclose{}= r \) and thus Φfunction⁡(0zero)=equals1one \( Φ\mathopen{}\left( 0\right)\mathclose{}= 1 \). As depicted in Figure B, as λnonnegative number\( λ \) increases, we find certain values force ffunctionλnonnegative number⁡(1one)=equals0zero \( {f}_{λ}\mathopen{}\left( 1\right)\mathclose{}= 0 \) .

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Figure A. Solutions to the initial value problem ffunction″second derivative⁡(rreal number)+plusλnonnegative number⁢timesrreal number⁢timesffunction⁡(rreal number)=equals0zero \( f'' \mathopen{}\left( r\right)\mathclose{}+λrf\mathopen{}\left( r\right)\mathclose{}= 0 \); ffunction⁡(0zero)=equals0zero \( f\mathopen{}\left( 0\right)\mathclose{}= 0 \), ffunction′derivative⁡(0zero)=equals1one \( f' \mathopen{}\left( 0\right)\mathclose{}= 1 \) for λnonnegative number\( λ \) equal to 0, 1, 7, and 19.
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Figure B. Φfunction⁡(λnonnegative number)=equalsffunctionλnonnegative number⁡(1one) \( Φ\mathopen{}\left( λ\right)\mathclose{}= {f}_{λ}\mathopen{}\left( 1\right)\mathclose{} \)
Call these λnonnegative number1one\( {λ}_{1} \), λnonnegative number2two\( {λ}_{2} \), and so on. We naturally ask if there are formulas for these values. No way! However, we will see in Section G that ∑summationjinteger=1one∞infinity1oneλnonnegative numberjinteger=equals∫integral0zero1one treal number⁢times(1one-minustreal number)⁢timesφcontinuous function⁡(treal number) ⁢dtreal number \[ \sum_{j=1}^{\infty}{}\frac{1}{{λ}_{j}}= \int _{0}^{1}{} t\mathopen{}\left(1-t\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t \] and ∑summationjinteger=1one∞infinity1oneλnonnegative numberjinteger2two=equals (∑summationjinteger=1one∞infinity1oneλnonnegative numberjinteger) 2two -minus2two⁢times∫integral0zero1one ∫integral0zerosreal number treal number⁢times(sreal number-minustreal number)⁢times(1one-minussreal number)⁢timesφcontinuous function⁡(treal number) ⁢dtreal number ⁢dsreal number \[ \sum_{j=1}^{\infty}{}\frac{1}{{{λ}_{j}}^{2}}= {\mathopen{}\left(\sum_{j=1}^{\infty}{}\frac{1}{{λ}_{j}}\right)\mathclose{}}^{2}-2\int _{0}^{1}{} \int _{0}^{s}{} t\mathopen{}\left(s-t\right)\mathclose{}\mathopen{}\left(1-s\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t \,\mathrm{d}s \] and so forth, in principle.

How we will eventually understand this: using the Hilbert space HHilbert space=equalsL2Lebesgue space⁡([interval0zero, 1one]intervalφcontinuous function⁡(treal number)⁢ ddifferentialtreal number) \( H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}, φ\mathopen{}\left( t\right)\mathclose{} \, \mathrm{d}t\right)\mathclose{} \) we define an operator Kcompact operator:mapsHHilbert space→toHHilbert space \( K : H \to H \) by (Kcompact operator⁡(ξfunction))⁡(sreal number)=equals−⁢ ∫integral0zerosreal number ξfunction⁡(treal number)⁢times(sreal number-minustreal number)⁢timesφcontinuous function⁡(treal number) ⁢dtreal number +plusxvector⁢times∫integral0zero1one ξfunction⁡(treal number)⁢times(1one-minustreal number)⁢timesφcontinuous function⁡(treal number) ⁢dtreal number . \[ \mathopen{}\left(K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( s\right)\mathclose{}= {-} \int _{0}^{s}{} ξ\mathopen{}\left( t\right)\mathclose{}\mathopen{}\left(s-t\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t +x\int _{0}^{1}{} ξ\mathopen{}\left( t\right)\mathclose{}\mathopen{}\left(1-t\right)\mathclose{}φ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t \text{.} \] The operator Kcompact operator\( K \) satisfies (Kcompact operator⁡(ξfunction))″second derivative+plusξfunction⁢timesφcontinuous function=equals0zero \( \mathopen{}\left(K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}'' +ξφ= 0 \) and is a positive compact operator on HHilbert space\( H \) with eigenvalues 1oneλcomplex number1one\( \frac{1}{{λ}_{1}} \), 1oneλcomplex number2two\( \frac{1}{{λ}_{2}} \), …. In fact, Kcompact operator\( K \) is trace-class (meaning the sum of the eigenvalues is finite) with trace ∑summationjinteger=1one∞infinity1oneλcomplex numberjinteger . \[ \sum_{j=1}^{\infty}{}\frac{1}{{λ}_{j}} \text{.} \]