Taylor’s formula in Banach spaces

Let U be an open subset of a real Banach spaceMathworldPlanetmath X. If f:U is differentiableMathworldPlanetmathPlanetmath n+1 times on U, it may be expanded by Taylor’s formula:

f(x)=f(a)+Df(a)h+12!D2f(a)h2++1n!Dnf(a)hn+Rn(x), (1)

with the following expressions for the remainder term Rn(x):

Rn(x) =1n!Dn+1f(η)(x-η)nh Cauchy form of remainder
Rn(x) =1(n+1)!Dn+1f(ξ)hn+1 Lagrange form of remainder
Rn(x) =1n!01Dn+1f(a+th)((1-t)h)nh𝑑t integral form of remainder

Here a and x must be points of U such that the line segment between a and x lie inside U, h is x-a, and the points ξ and η lie on the same line segment, strictly between a and x.

The kth Fréchet derivative of f at a is being denoted by Dkf(a), to be viewed as a multilinear map Xk. The hk notation means to evaluate a multilinear map at (h,,h).

1 Remainders for vector-valued functions

If Y is a Banach space, we may also consider Taylor expansionsMathworldPlanetmath for f:UY. Formula (4) takes the same form, but the Cauchy and Lagrange forms of the remainder will not be exact; they will only be bounds on Rn(x). That is, for f:UY,

Rn(x) 1n!Dn+1f(η)(x-η)nh Cauchy form of remainder
Rn(x) 1(n+1)!Dn+1f(ξ)hn+1 Lagrange form of remainder

It is not hard to find counterexamplesMathworldPlanetmath if we attempt to remove the norm signs or change the inequalityMathworldPlanetmath to equality in the above formulas.

However, the integral form of the remainder continues to hold for Y, although strictly speaking it only applies if the integrand is integrable. The integral form is also applicable when X and Y are complex Banach spaces.

Mean Value Theorem

The Mean Value Theorem can be obtained as the special case n=0 with the Lagrange form of the remainder: for f:UY differentiable,

f(x)-f(a)Df(ξ)(x-a) (2)

If Y=, then the norm signs may be removed from (2), and the inequality replaced by equality.

Formula (2) also holds under the much weaker hypothesis that f only has a directional derivative along the line segment from a to x.

Weaker bounds for the remainder

If f:UY is only differentiable n times at a, then we cannot quantify the remainder by the n+1th derivative, but it is still true that

Rn(x)=o(x-an) as xa (3)

Finite-dimensional case

If X=m and Y=, Dk has the following expression in terms of coordinates:


where each ij runs from 1,,m in the sum.

If we collect the equal mixed partials (assuming that they are continuousMathworldPlanetmath) then


where J is a multi-index of m components, and each component Ji indicates how many times the derivative with respect to the ith coordinate should be taken, and the exponentPlanetmathPlanetmath that the ith coordinate of h should be raised to in the monomial hJ. The multi-index J runs through all combinationsMathworldPlanetmath such that J1++Jm=|J|=k in the sum. The notation J! means J1!Jm!.

All this is more easily assimilated if we remember that Dkf(a)hk is supposed to be a polynomial of degree k. Also |J|!/J! is just the multinomial coefficientMathworldPlanetmath.

Taylor series

If limnRn(x)=0, then we may write

f(x)=f(a)+Df(a)h+12!D2f(a)h2+ (4)

as a convergent infinite series. Elegant as such an expansion is, it is not seen very often, for the reason that higher order Fréchet derivatives, especially in infinite-dimensional spaces, are often difficult to calculate.

But a notable exception occurs if a function f is defined by a convergent “power seriesMathworldPlanetmath

f(x)=k=0Mk(x-a)k (5)

where {Mk:k=0,1,} is a family of continuous symmetric multilinear functions XkY. In this case, the series (5) is the Taylor series for f at a.


  • 1 Arthur Wouk. A course of applied functional analysisMathworldPlanetmathPlanetmath. Wiley-Interscience, 1979.
  • 2 Eberhard Zeidler. Applied functional analysis: main principles and their applications. Springer-Verlag, 1995.
  • 3 Michael Spivak. Calculus, third edition. Publish or Perish, 1994.
Title Taylor’s formula in Banach spaces
Canonical name TaylorsFormulaInBanachSpaces
Date of creation 2013-03-22 15:28:27
Last modified on 2013-03-22 15:28:27
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 10
Author stevecheng (10074)
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Classification msc 46T20
Classification msc 26B12
Classification msc 41A58