An introductory textbook on functional analysis and operator theory I would like to ask for some recommendation of introductory texts on functional analysis. I am not a professional mathematician and I am totally new to the subject. However, I found out that some knowledge of functional analysis and operator theory would be quite helpful to my work...
What I am searching for is some accessible and instructive text not necessarily covering the subject in great depth, but explaining the main ideas. I am not searching for a text for engineers, some amount of mathematical rigor would be fine. But I found myself unable of reading some standard textbooks covering in great depth a large amount of issues in theory of Banach spaces, etc. I am looking for something that proceeds to the most important topics (e.g., spectral theory) faster than the most of textbooks, but not at the expense of rigor. I.e., something that covers rigorously the main topics, but concentrates only on the main ideas.
Simply an accessible introductory text for a fast orientation in the subject.
Moreover, I would prefer a text that does not require any background in measure theory and similar disciplines.
And another question: is there any functional analysis book that deals primarily with sequence spaces? It need not fulfill the description above.
Thank you for your recommendations! 
 A: Bollobas's Linear Analysis is short and rigorous but may be slightly too compressed (depending on your background). The second half of Simmons's Topology and Modern Analysis remains, in my view, one of the best "soft" introductions to Banach algebras, the spectrum, and ultimately the Gelfand Naimark theorem for commutative C^*-algebras.
If you don't mind just dealing with Hilbert spaces, then the first two thirds of Young's Introduction to Hilbert Spaces may also be worth a look.
None of these texts require any measure theory.
A: The best introductory text I know on the subject is available from Dover books very cheap; it's George Bachman and Lawrence Narici's Functional Analysis. It develops virtually the entire subject of functional analysis from scratch with many examples and good exercises-and it requires only advanced calculus (elementary real analysis) and a good working knowledge of linear algebra to work through. 
Also, if you just "need to know some functional analysis" for work, you might want to check out some applied textbooks on the subject, of which there are many. 
A: I think that Kreyszig'book is a good introduction, though not very short.
Spaces of sequences are classical Banach spaces, and there are books that study their properties systematically. A classic book is this one. But also Larsen's book has plenty of examples from $\ell^p$, $c_0$, $c_{00}$.
