This book is meant as a text for a first-year graduate course in analysis. In a sense, it covers the same topics as elementary calculus but treats them in a manner suitable for people who will be using it in further mathematical investigations. The organization avoids long chains of logical interdependence, so that chapters are mostly independent. This allows a course to omit material from some chapters without compromising the exposition of material from later chapters.
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11 of 11 people found the following review helpful:
5.0 out of 5 stars
One of a kind !,
This review is from: Real and Functional Analysis (Graduate Texts in Mathematics) (v. 142) (Hardcover)
Up to my knowledge, this is the only book that constructs the Lebesgue integral for functions to a general Banach-space instead of the real numbers (thus saving us from the unnecessary and esthetically dissapointing construction through positive and negative functions).
I don't know how Lang does it, but eerytime you'll pick up one of his books, you'll marvel at the beauty of mathematics !
13 of 14 people found the following review helpful:
5.0 out of 5 stars
Much better than Royden!,
This review is from: Real and Functional Analysis (Graduate Texts in Mathematics) (v. 142) (Hardcover)
It drove me up the wall, in my first course on measure and integration, that integration was first done for positive functions, then for real functions by writing them as a difference of positive functions, then complex functions in terms of real and imaginary parts. Why couldn't you just integrate real-valued functions
intrinsically, without the silly decomposition into positive and negative parts? After that course, I found Lang's book. What a blessing to see that you can just integrate in infinite-dimensional spaces right from the start. I can't understand why virtually all books on integration theory still succumb to the "positive functions first" approach.
7 of 7 people found the following review helpful:
4.0 out of 5 stars
Overall, A Good Book,
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This review is from: Real and Functional Analysis (Graduate Texts in Mathematics) (v. 142) (Hardcover)
I've read several analysis books and this is one of the better ones that I have read. It covers a variety of interesting and useful topics and the exposition is clear. It's presentation is a bit more abstract than some others starting with some functional-analytic concepts before doing integration in that framework. However, if you want to study stochastic analysis, getting in this frame of mind will definitely help your understanding of stochastic integration. For a truly thorough understaning of the subject, I recommend purchasing this book as well as the somewhat easier "Lebesgue Integration on Euclidean Space" by Frank Jones - the two together cost about the same as Royden, Rudin, or the terrible book by Aliprantis.
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Inside This Book
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First Sentence:
We assume that the reader understands the meaning of the word "set", and in this chapter, summarize briefly the basic properties of sets and operations between sets. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
toplinear isomorphism, linear extension theorem, positive hermitian operators, shrinking map, shrinking lemma, proves our corollary, modulo compact operators, block spanned, normed vector space, continuous bilinear map, inverse mapping theorem, denumerable union, spectral family, proves our lemma, continuous linear map, step maps, proves our theorem, admissible subset, disjoint measurable sets, hermitian product, open mapping theorem, sup norm, averaging theorem, finite measure, normed algebra Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Undergraduate Analysis New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
We assume that the reader understands the meaning of the word "set", and in this chapter, summarize briefly the basic properties of sets and operations between sets. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
toplinear isomorphism, linear extension theorem, positive hermitian operators, shrinking map, shrinking lemma, proves our corollary, modulo compact operators, block spanned, normed vector space, continuous bilinear map, inverse mapping theorem, denumerable union, spectral family, proves our lemma, continuous linear map, step maps, proves our theorem, admissible subset, disjoint measurable sets, hermitian product, open mapping theorem, sup norm, averaging theorem, finite measure, normed algebra Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Undergraduate Analysis New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
100
books
cite this book:
See all 100 books citing this book
- All the Mathematics You Missed: But Need to Know for Graduate School by Thomas A. Garrity on page 43, and Back Matter
- Complex Analysis (Chapman & Hall/CRC Pure and Applied Mathematics) by Mario Gonzalez in Front Matter
- Functions of One Complex Variable (Graduate Texts in Mathematics - Vol 11) (v. 1) by John B Conway in Back Matter
- An Introduction to Number Theory (Graduate Texts in Mathematics) by G. Everest in Back Matter
- Geometric Function Theory in One and Higher Dimensions (Pure and Applied Mathematics, Vol 255) (Chapman & Hall/CRC Pure and Applied Mathematics) by Ian Graham in Front Matter
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