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In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As an undergrad who feels like at times he knows more about math then the actual rigor and computation involved, are there any good recommendations to an introductory functional analysis book that give a reasonable treatment as well as show some connections it may have to other fields of math?

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So what is your background? How much real analysis? Any measure theory? –  Michael Greinecker Apr 7 at 15:47
    
In my opinion (and Halmos') the best way to study Functional Analysis is by a problem solving approach. Halmos' A Hilbert space problem book may not be easy (or appropriate for beginning). My suggestion is "Trenoguin's Problems and exercises in Functional Analysis. Mir." –  user141267 Apr 7 at 15:51
    
@MichaelGreinecker In regards to measure, my background doesn't go much past the definition of measure and the Baire-Category theorem, which we touched upon in my topology course. –  Just Some Math Major Apr 7 at 16:06
    
Introduction to Topology and Modern Analysis by George F. Simmons is a possibility. –  Dave L. Renfro Apr 7 at 17:04

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One of the few books I've seen that is fully self-contained at the undergraduate level is available from Dover: Functional Analysis by George Bachman and Lawrence Narici. The exposition is verbose and unusually well-written; so don't be too put off by the length. You can teach yourself the subject from this book. http://www.amazon.com/Functional-Analysis-Dover-Books-Mathematics/dp/0486402517/ref=pd_sxp_grid_i_1_0

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A gentle path that brings you relatively far is to work through Introduction to the Analysis of Metric Spaces by John Giles and then through his Introduction to the Analysis of Normed Linear Spaces. In spite of its name, the first book has a lot of material on normed spaces.

A nice book with an applied outlook is Optimization by Vector Space Methods by David Luenberger.

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