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I am new to topology & vector spaces with rudimentary knowledge. Could any one suggest me a comprehensive book on topology with vector spaces ? I find topology & vector spaces - mathematical tool which best suites my research work.

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Do you have any specific applications in mind? What are you learning this for? –  Michael Greinecker Dec 29 '12 at 12:00
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It would be helpful if you gave an overview of what you do know about topology and vector spaces. –  Jay Dec 29 '12 at 13:01
    
I recommend Munkres' book for studying topology. –  Fang Jing May 13 at 17:18

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In the absence of further information about the required level, here are my suggestions.

If you are really interested in topological vector spaces (not sure from the question, this might be more ambitious than you intended), then perhaps you might take a look at these online lecture notes by I. F. Wilde: Basic Analysis – Gently Done (Topological Vector Spaces).

If, on the other hand you need recommendations for topology and vector spaces separately, then the simplest thing to do is a search on Amazon for "Introduction to vector spaces/linear algebra/topology" and see what comments have been left by other purchasers. My personal preference would be for "Introduction to Topology and Modern Analysis" (G. F. Simmons) which is a very readable treatment of topology which also covers some vector space stuff in a chapter on "Algebraic Systems".

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+1. I downloaded Wilde's but haven't really read it. –  Tim Dec 29 '12 at 16:57
    
@Tim I have only read parts of it, but I liked his treatment of nets and of TVSs, but a pity there are no exercises. –  Old John Dec 29 '12 at 17:01
    
I tend to think having no exercise usually tends to be more self-contained, so it is not a pity for me –  Tim Dec 29 '12 at 17:04

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