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I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be appreciated.

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Could you be a little bit more explicit? Are you talking about this functional analysis? Or do you want to learn calculus of variations (for functionals)? Or do you want to learn to functional differentiation and integration. Or perhaps something else altogether? –  Marek Mar 29 '11 at 6:54
    
Just to add to Marek's comment. Or do you mean that part of functional analysis within the context of the spectral theorem, the way to evaluate functions at operators? See en.wikipedia.org/wiki/Functional_calculus –  MBN Mar 29 '11 at 12:52
    
Just edited the post. Sorry for the ambiguous. I am not a native western language speaker, not educated in western world, and didn't take any math course other than 1st yr college calculus (I can do calculus as well as 12 yr old Jake, though). So I ask your pardon for not knowing the correct terminology. Or only those good at English are welcomed to this forum? –  skywaddler Mar 29 '11 at 15:10
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@skywaddler. Sorry, but I got to ask for clarification: In your terminology, do you distinguish between a function and a functional ? –  Qmechanic Mar 29 '11 at 17:41
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Dear @skywaddler, At the time when I wrote my above comment, your formulation of the question was ambiguous. Please trust me when I state that the comment was genuinely meant to clarify your question, and nothing else. –  Qmechanic Mar 30 '11 at 14:18
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2 Answers

Have a look first at several chapters in Stone and Goldbart, "Mathematics for Physics" (the free preprint is here) before entering into more specific books. I think you may want to see chapters 1, and parts of 2 and 9.

You may find some parts of what you want in classic books of the "Comprehensive Mathematical Methods for Physics" type, but they don't usually cover that questions in detail. Stone&Goldbart, without being a dedicated book, is somewhere in between.

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The standard encyclopedic treatise of nonlinear functional analysis is the 5 volume opus of Eberhard Zeidler, "Nonlinear Functional Analysis and Its Applications". It covers a lot of material about variational calculus, for example, in volume III "Variational Methods and Optimization". The applications are usually applications from physics.

If that is too much material, there is also a two volume version including some topics of linear functional analysis as well, "Applied functional analysis. Main principles and their applications." and "Applied functional analysis. Applications to mathematical physics."

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