Taylor’s formula in Banach spaces
Let $U$ be an open subset of a real Banach space^{} $X$. If $f:U\to \mathbb{R}$ is differentiable^{} $n+1$ times on $U$, it may be expanded by Taylor’s formula:
$$f(x)=f(a)+\mathrm{D}f(a)\cdot h+\frac{1}{2!}{\mathrm{D}}^{2}f(a)\cdot {h}^{2}+\mathrm{\cdots}+\frac{1}{n!}{\mathrm{D}}^{n}f(a)\cdot {h}^{n}+{R}_{n}(x),$$ | (1) |
with the following expressions for the remainder term ${R}_{n}(x)$:
${R}_{n}(x)$ | $={\displaystyle \frac{1}{n!}}{\mathrm{D}}^{n+1}f(\eta )\cdot {(x-\eta )}^{n}h$ | Cauchy form of remainder | ||
${R}_{n}(x)$ | $={\displaystyle \frac{1}{(n+1)!}}{\mathrm{D}}^{n+1}f(\xi )\cdot {h}^{n+1}$ | Lagrange form of remainder | ||
${R}_{n}(x)$ | $={\displaystyle \frac{1}{n!}}{\displaystyle {\int}_{0}^{1}}{\mathrm{D}}^{n+1}f(a+th)\cdot {((1-t)h)}^{n}h\mathit{d}t$ | 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 $\xi $ and $\eta $ lie on the same line segment, strictly between $a$ and $x$.
The $k$th Fréchet derivative of $f$ at $a$ is being denoted by ${\mathrm{D}}^{k}f(a)$, to be viewed as a multilinear map ${X}^{k}\to \mathbb{R}$. The $\cdot {h}^{k}$ notation means to evaluate a multilinear map at $(h,\mathrm{\dots},h)$.
1 Remainders for vector-valued functions
If $Y$ is a Banach space, we may also consider Taylor expansions^{} for $f:U\to Y$. 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 ${R}_{n}(x)$. That is, for $f:U\to Y$,
$\parallel {R}_{n}(x)\parallel $ | $\le {\displaystyle \frac{1}{n!}}\parallel {\mathrm{D}}^{n+1}f(\eta )\cdot {(x-\eta )}^{n}h\parallel $ | Cauchy form of remainder | ||
$\parallel {R}_{n}(x)\parallel $ | $\le {\displaystyle \frac{1}{(n+1)!}}\parallel {\mathrm{D}}^{n+1}f(\xi )\cdot {h}^{n+1}\parallel $ | Lagrange form of remainder |
It is not hard to find counterexamples^{} if we attempt to remove the norm signs or change the inequality^{} to equality in the above formulas.
However, the integral form of the remainder continues to hold for $Y\ne \mathbb{R}$, 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 $f:U\to Y$ differentiable,
$$\parallel f(x)-f(a)\parallel \le \parallel \mathrm{D}f(\xi )\cdot (x-a)\parallel $$ | (2) |
If $Y=\mathbb{R}$, 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 $f:U\to Y$ is only differentiable $n$ times at $a$, then we cannot quantify the remainder by the $n+1$th derivative, but it is still true that
$${R}_{n}(x)=o({\parallel x-a\parallel}^{n})\text{as}x\to a\text{.}$$ | (3) |
Finite-dimensional case
If $X={\mathbb{R}}^{m}$ and $Y=\mathbb{R}$, ${\mathrm{D}}^{k}$ has the following expression in terms of coordinates:
$${\mathrm{D}}^{k}f(a)\cdot ({\xi}_{1},\mathrm{\dots},{\xi}_{k})=\sum _{{i}_{1},\mathrm{\dots},{i}_{k}}\frac{{\partial}^{k}f}{\partial {x}^{{i}_{1}}\mathrm{\cdots}\partial {x}^{{i}_{k}}}{\xi}_{1}^{{i}_{1}}\mathrm{\cdots}{\xi}_{k}^{{i}_{k}},$$ |
where each ${i}_{j}$ runs from $1,\mathrm{\dots},m$ in the sum.
If we collect the equal mixed partials (assuming that they are continuous^{}) then
$$\frac{1}{k!}{\mathrm{D}}^{k}f(a)\cdot {h}^{k}=\sum _{|J|=k}\frac{1}{J!}\frac{{\partial}^{|J|}f}{\partial {x}^{J}}{h}^{J},$$ |
where $J$ is a multi-index of $m$ components, and each component ${J}_{i}$ indicates how many times the derivative with respect to the $i$th coordinate should be taken, and the exponent^{} that the $i$th coordinate of $h$ should be raised to in the monomial ${h}^{J}$. The multi-index $J$ runs through all combinations^{} such that ${J}_{1}+\mathrm{\cdots}+{J}_{m}=|J|=k$ in the sum. The notation $J!$ means ${J}_{1}!\mathrm{\cdots}{J}_{m}!$.
All this is more easily assimilated if we remember that ${\mathrm{D}}^{k}f(a)\cdot {h}^{k}$ is supposed to be a polynomial of degree $k$. Also $|J|!/J!$ is just the multinomial coefficient^{}.
Taylor series
If ${lim}_{n\to \mathrm{\infty}}{R}_{n}(x)=0$, then we may write
$$f(x)=f(a)+\mathrm{D}f(a)\cdot h+\frac{1}{2!}{\mathrm{D}}^{2}f(a)\cdot {h}^{2}+\mathrm{\cdots}$$ | (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 series^{}”
$$f(x)=\sum _{k=0}^{\mathrm{\infty}}{M}_{k}\cdot {(x-a)}^{k}$$ | (5) |
where $\{{M}_{k}:k=0,1,\mathrm{\dots}\}$ is a family of continuous symmetric multilinear functions ${X}^{k}\to Y$. In this case, the series (5) is the Taylor series for $f$ at $a$.
References
- 1 Arthur Wouk. A course of applied functional analysis^{}. 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) |
Entry type | Result |
Classification | msc 46T20 |
Classification | msc 26B12 |
Classification | msc 41A58 |