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I am writing this post because I would like to know what are your references concerning math book showing the interplay between analysis and algebra at an undergraduate-advanced undergraduate level.

Most of math textbooks treat either analysis or algebra (and linear algebra) but it is not usual to see a unified treatment of these two areas of mathematics.

And I think that's a pity, because it's a vision that's necessary for almost all advanced areas of mathematics (just to mention functional analysis, for example...).

I know of a few books on multivariate calculus and linear algebra, but most don't take a rigorous approach to the subject.

The only potential book that I found is « Analysis in vector space » by Mustafa A. Akcoglu et al. but it is not available in my library and very expensive to buy it.

Thank you a lot !

I think that with time and practice you get acquainted to this « big picture » but since I am self study math for the most part, a book is very welcome.

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    $\begingroup$ Nice question. Whenever possible, students should learn how different areas of mathematics are related in non-trivial ways. $\endgroup$
    – jg1896
    Commented Nov 17, 2023 at 14:55
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    $\begingroup$ Your best bet is to read separate texts on algebra and analysis (ideally well-written by experts in the respective subjects, and chosen to fit your prerequisites), and get some insight on the connections from Evan Chen's Napkin. The linear-algebra-analysis combination is somewhat more popular (e.g., Olver and Shakiban exists), but abstract-algebra-analysis is rare due to the lack of writers with the appropriate competence and interest on both sides. $\endgroup$
    – darij grinberg
    Commented Nov 17, 2023 at 15:13
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    $\begingroup$ (17 hours old comment reposted because last word "quality" should be "qualify") Advanced Calculus by Nickerson/Spencer/Steenrod (1959 original publication and 2011 Dover reprint) has quite a bit of algebra in it, although a very generous interpretation of "undergraduate-advanced undergraduate level" is needed for it to qualify. $\endgroup$
    – Dave L Renfro
    Commented Nov 18, 2023 at 12:34
  • $\begingroup$ @DaveLRenfro this is a great suggestion. Another book roughly the same topic I like very much is Loomis. Sternberg, Advanced Calculos. $\endgroup$
    – jg1896
    Commented Nov 18, 2023 at 13:39
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    $\begingroup$ @jg1896: I was thinking of that book also (I've known about both books since around 1977), but I only mentioned Nickerson/Spencer/Steenrod because in both cases the "algebra" is almost entirely linear algebra and I also felt that Nickerson/Spencer/Steenrod has more algebraic character. Incidentally, most everywhere online that mentions reviews about this book (maybe everywhere) cite and/or excerpt from the very enthusiastic review in Bull. AMS (to a lesser extent, the review in Amer. Math. Monthly). But see Philip J. Davis's review in Scripta Mathematica 25 #3 (November 1960), pp. 258-259. $\endgroup$
    – Dave L Renfro
    Commented Nov 18, 2023 at 18:59

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I can recommend two books.

The first one is Simmon's 'Introduction to topology and modern analysis'.

I myself studied some portions of this book when an undergraduate.It has three parts. Part one is a very concise introduction to the main parts of general topology. Part two is on operators on normed vector spaces, in particular Banach and Hilbert spaces. Part three is an introduction to operator algebras, and has many important results such as Gelfand-Neumark Theorem and Banach-Stone Theorem.

I also would say it writing style is very clear.

A second book, which is a little more oriented to research but is acessible to advanced undergraduates is Coutinho's 'A primer of algebraic D-modules'.

This is one of most well written books I known of.

It discusses the extremely important theory of D-modules in the concrete case of the Weyl algebra. It is mainly an algebra book, but it presents lots of applications of the theory to analysis, in particular, differential equations, and it introduces the fundamental notion of holonomic modules. This book also discusses some symplectic geometry, when dealing with the characteristic variety. It finishes with a elementary introduction to a fundamental result, Kashiwara's Theorem.

I hope this recommendations are among the lines of what you seek!

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    $\begingroup$ Courant and Hilbert Methods of Mathematical Physics vol I. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 17, 2023 at 17:38

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