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Hi, I want to know difference that between 'functional' and 'function'.

Of course, in Wikipedia, http://en.wikipedia.org/wiki/Functional_(mathematics), there is many texts.

But what's the simple answer for this question? ;-)

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closed as off topic by Qiaochu Yuan, José Figueroa-O'Farrill, Loop Space, Charles Siegel, Dan Petersen May 24 '11 at 14:16

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This question is a bit too basic to be appropriate here, as this site is intended for "research-level mathematics". Please see the FAQ: mathoverflow.net/faq. You may get better answers and a warmer welcome on math.stackexchange.com or any of the alternatives mentioned in the FAQ. –  Theo Buehler May 24 '11 at 8:49
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when you start working with things like "a function that takes functions and returns functions", you realize immediately that you need some alternate terminology in order to preserve your sanity. That's where terms like operator and functional come from. –  Federico Poloni May 24 '11 at 11:00
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Just imagine you encounter a function defined on a space of functionals. Would you call that a functionalal? –  Konrad Waldorf May 24 '11 at 11:27
    
@Konrad: ha ha, that's exactly what my friend James Dolan would call it (but he is known for being a bit eccentric at times). –  Todd Trimble May 25 '11 at 13:47
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1 Answer 1

No difference really, just convention.

A functional normally acts on other functions, whereas a function normally acts on some underlying vector space or field. But this is not always true.

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I always thought a functional was a mapping from a vector space to its field of scalars. That's the definition in Rudin's text on functional analysis, anyway. –  Simon Lyons May 24 '11 at 10:40
    
I think this definition is also somewhere in Lang's Linear Algebra. –  Giuseppe Tortorella Jun 24 '11 at 11:46
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