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may I know what is the distinction between functional analysis and linear functional analysis? I do a search online and came to a conclusion that linear functional analysis is not functional analysis and am getting confused by them. When I look at the book Linear Functional Analysis published by Joan Cerda by AMS, there is not much difference in its content compared to other titles with just the heading Functional Analysis. Hope someone can clarify for me. Thank You.

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up vote 7 down vote accepted

It is mostly a matter of taste. Traditionally functional analysis has dealt mostly with linear operators, whereas authors would typically title their books nonlinear functional analysis if they consider the theory of differentiable (or continuous or more general) mappings between Banach (or more general function) spaces.

But it seems more recently the subject of "nonlinear functional analysis" has been gaining traction and many more people now adhere to the philosophy "classifying mathematical problems as linear and nonlinear is like classifying the universe as bananas and non-bananas." So instead of labeling all the non-bananas, some people now agree that it makes more sense to instead label the bananas...

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They probably mean the same thing. It's a matter of emphasis. Cerda may be trying to distinguish ordinary functional analysis from nonlinear functional analysis, where nonlinear maps are studied.

Bollobás does something similar; his book on functional analysis is entitled Linear Analysis and I think this is again to emphasize the linear nature of the subject.

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