Functionals->functions of infinite variables?

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Functionals can be viewed as functions of an infinite number of Taylor coefficients, suggesting they may resemble normal functions that map sets of reals to other sets of reals. However, not all variable functions possess a Taylor expansion, and even if they do, the expansion may only be valid in certain domains. This raises questions about the nature of functionals, especially in spaces like L2, which includes discontinuous functions and those lacking derivatives. The discussion highlights the complexities of defining functionals in relation to traditional functions. Ultimately, the relationship between functionals and functions remains nuanced and context-dependent.
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If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.
 
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HomogenousCow said:
If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.

The "variable function" may not have a Taylor expansion, or if it does for part of its domain, the expansion may not hold for the rest. Simple example, L2 includes discontinuous functions, functions without derivatives, etc.
 

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