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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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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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