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?
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
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.