My Math Forum Series expansion of a functional around a function

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 January 15th, 2010, 12:38 PM #1 Newbie   Joined: Dec 2009 Posts: 2 Thanks: 0 Series expansion of a functional around a function Hello, I am looking for a general formula for the expansion of a functional around a function. I am wondering if anyone knows what book or reference I can look at to learn this topic. I've looked everywhere and the closest thing I've found was on page 234 of this pdf file (It will say pg 234 in acrobat reader, but pg 226 on the actual page since the first few pages are not numbered): http://www.math.uwaterloo.ca/~mscott/Notes.pdf The funcitonal Taylor series is defined under the word "Proof" , but only for the very special case of x(t)=0. I'm looking for the general expression for this formula, that is not specific to the case x(t)=0. I hope there is a book or reference that has this formula that explains it becuase I have no clue where those integrals and multiple functional derivatives are coming from. Thank you
 January 16th, 2010, 02:01 PM #2 Member     Joined: Aug 2007 Posts: 42 Thanks: 4 Re: Series expansion of a functional around a function It's only an analog of the Taylor expansion formula. For a smooth functional F on a Banach space U, it reads $F(u)=F(u_0)+F'(u_0)(u-u_0)+\frac{1}{2}(F'#39;(u_0)(u-u_0,u-u_0))+..., \ u,u_0\in U$. where $F',F''$ are the corresponding order Frechet derivatives.

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