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Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.

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    $\begingroup$ I seem to remember Spivak (Calculus) having plenty of diagrams. If that's too low level, try Pugh's Real Mathematical Analysis. At some point the geometry is going to have to be behind the scenes, though... $\varepsilon$ and $\delta$ are the stars of the show in real analysis. I don't have any strictly topological suggestions, sorry. $\endgroup$ – Tyler Jul 7 '14 at 19:51
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Probably too old-fashioned, but a reference which has a some decent geometric insights (and has the advantage of being legally free on Gutenberg) is:

G. H. Hardy: A Course in Pure Mathematics.

It looks rather dated now, but there still a few of us here who used it as the basis for our understanding of analysis.

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My favorite textbooks on analysis with many pictures are:

1) Vladimir A. Zorich "Mathematical Analysis I & II"

This is a slow-paced introduction to real analysis, starting from the concept of real number and ending at analysis on manifolds, Fourier series and asymptotic methods. The author uses many complementary disciplines to adequately place real analysis in modern mathematics curriculum: he uses metric and general topology to discuss notions of continuity and limit, notions from linear algebra to give a clear picture of what differential is in multivariable setting. The book also has many exercises, often problems discuss various physics applications of real analysis.

The only drawback is that the textbook doesn't have a chapter on Lebesgue integration, everything is done in Riemann setting.

2) Charles C. Pugh "Real Mathematical Analysis"

This textbook is much shorter and many important facts are not included in the text, but are asked to be proved in the exercises (500+ of them). However, the book is full of geometric insights and it does have a chapter on Lebesgue integration.

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"Yet Another Introduction to Analysis" by Victor Bryant.

The book gets a lot of "slack" and is often said to not be a rigorous text, but if your looking to learn analysis for the first time and are fairly new to the subject, I think the book is wonderful. The most terse text in the subject is not always the best.

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