Re: Functional Analysis: Equivalent of Taylor series for operators

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 12/01/04 

Date: Wed, 01 Dec 2004 05:13:52 -0600

On 1 Dec 2004 00:09:23 GMT, israel@math.ubc.ca (Robert Israel) wrote:

>In article <v7soq0t2aoc7l17s4ndmvv09igj1ut4n4r@4ax.com>,
>David C. Ullrich <ullrich@math.okstate.edu> wrote:
>>On 30 Nov 2004 00:16:09 GMT, israel@math.ubc.ca (Robert Israel) wrote:
>
>>>Taylor's theorem for functions from a Banach space to a Banach space
>>>is Theorem 6 in Nelson, "Topics in Dynamics I: Flows", Princeton U. Press
>>>1969.
>
>>Huh. Can you give us a hint what sort of thing these are? I can
>>only imagine two possibilities: Taylor series which really
>>amount to series for the function of one variable
>>f(t) = T(x + ty), or something like Taylor series in infinitely
>>many variables, by analogy with Taylor series in R^n.
>
>For a function f from an open subset U of Banach space E to Banach space
>F, the Frechet derivative (if it exists) is a function Df from U to
>L(E,F) (the Banach space of bounded linear operators E -> F) such that
>f(x+y) = f(x) + Df(x) y + o(||y||). The Frechet derivative of Df, if it
>exists, is then a function D^2 f from U to L(E,L(E,F)),

Realized a minute too late that this was probably what was
going on. Duh. Thanks.

>which can be
>identified with the space L(E x E, F) of bounded bilinear forms from
>E x E to F. We write D^2 f(x) y^2 for the value of the form (D^2 f)(x)
>at (y,y). Similarly for any positive integer k, D^k f, if it exists, is a
>function from U to the bounded k-linear forms from E^k to F, and we write
>D^k f(x) y^k for (D^k f)(x)(y,...,y) with k y's. Then this
>version of Taylor's theorem says that if f is C^k on U
>
>f(x+y) = f(x) + sum_{j=1}^k D^j f(x) y^j/j! + o(||y||^k)
>
>Of course if E is R^n and F is R, this can be thought of as a
>coordinate-free way of writing the multivariate Taylor series
>of f to k'th order.
>
>Robert Israel israel@math.ubc.ca
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia Vancouver, BC, Canada

************************

David C. Ullrich

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