Hadamard's lemma

{{math theorem|name=Hadamard's lemma{{sfn|Nestruev|2020|pp=17-18}}|note=|style=|math_statement=

Let $f$ be a smooth, real-valued function defined on an open, star-convex neighborhood $U$ of a point $a$ in $n$-dimensional Euclidean space. Then $f(x)$ can be expressed, for all $x\; \backslash in\; U,$ in the form:

$$f(x)\; =\; f(a)\; +\; \backslash sum\_\{i=1\}^n\; \backslash left(x\_i\; -\; a\_i\backslash right)\; g\_i(x),$$

where each $g\_i$ is a smooth function on $U,$ $a\; =\; \backslash left(a\_1,\; \backslash ldots,\; a\_n\backslash right),$ and $x\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right).$

}}

{{math proof|drop=hidden|proof=

Let $x\; \backslash in\; U.$ Define $h\; :\; [0,\; 1]\; \backslash to\; \backslash R$ by

$$h(t)\; =\; f(a\; +\; t(x\; -\; a))\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; t\; \backslash in\; [0,\; 1].$$

Then

$$h\text{'}(t)\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}(a\; +\; t(x\; -\; a))\; \backslash left(x\_i\; -\; a\_i\backslash right),$$

which implies

&= \int_0^1 \sum_{i=1}^n \frac{\partial f}{\partial x_i}(a + t(x - a)) \left(x_i - a_i\right)\, dt\\

&= \sum_{i=1}^n \left(x_i - a_i\right)\int_0^1 \frac{\partial f}{\partial x_i}(a + t(x - a))\, dt.\end{aligned}

But additionally, $h(1)\; -\; h(0)\; =\; f(x)\; -\; f(a),$ so by letting

$$g\_i(x)\; =\; \backslash int\_0^1\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}(a\; +\; t(x\; -\; a))\backslash ,\; dt,$$

the theorem has been proven.

$\backslash blacksquare$

}}

{{math theorem|name=Corollary{{sfn|Nestruev|2020|pp=17-18}}|note=|style=|math_statement=

If $f\; :\; \backslash R\; \backslash to\; \backslash R$ is smooth and $f(0)\; =\; 0$ then $f(x)/x$ is a smooth function on $\backslash R.$

Explicitly, this conclusion means that the function $\backslash R\; \backslash to\; \backslash R$ that sends $x\; \backslash in\; \backslash R$ to

$$\backslash begin\{cases\}$$

f(x)/x & \text{ if } x \neq 0 \\

\lim_{t \to 0} f(t)/t & \text{ if } x = 0 \\

\end{cases}

is a well-defined smooth function on $\backslash R.$

}}

{{math proof|drop=hidden|proof=

By Hadamard's lemma, there exists some $g\; \backslash in\; C^\{\backslash infty\}(\backslash R)$ such that $f(x)\; =\; f(0)\; +\; x\; g(x)$ so that $f(0)\; =\; 0$ implies $f(x)/x\; =\; g(x).$

$\backslash blacksquare$

}}

{{math theorem|name=Corollary{{sfn|Nestruev|2020|pp=17-18}}|note=|style=|math_statement=

If $y,\; z\; \backslash in\; \backslash R^n$ are distinct points and $f\; :\; \backslash R^n\; \backslash to\; \backslash R$ is a smooth function that satisfies $f(z)\; =\; 0\; =\; f(y)$ then there exist smooth functions $g\_i,\; h\_i\; \backslash in\; C^\{\backslash infty\}\backslash left(\backslash R^n\backslash right)$ ($i\; =\; 1,\; \backslash ldots,\; 3n\; -\; 2$) satisfying $g\_i(z)\; =\; 0\; =\; h\_i(y)$ for every $i$ such that

$$f\; =\; \backslash sum\_\{i\}^\{\}\; g\_i\; h\_i.$$

}}

{{math proof|drop=hidden|proof=

By applying an invertible affine linear change in coordinates, it may be assumed without loss of generality that $z\; =\; (0,\; \backslash ldots,\; 0)$ and $y\; =\; (0,\; \backslash ldots,\; 0,\; 1).$

By Hadamard's lemma, there exist $g\_1,\; \backslash ldots,\; g\_n\; \backslash in\; C^\{\backslash infty\}\backslash left(\backslash R^n\backslash right)$ such that

$f(x)\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; g\_i(x).$

For every $i\; =\; 1,\; \backslash ldots,\; n,$ let $\backslash alpha\_i\; :=\; g\_i(y)$ where $0\; =\; f(y)\; =\; \backslash sum\_\{i=1\}^n\; y\_i\; g\_i(y)\; =\; g\_n(y)$ implies $\backslash alpha\_n\; =\; 0.$

Then for any $x\; =\; \backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; \backslash in\; \backslash R^n,$

$$\backslash begin\{alignat\}\{8\}$$

f(x)

&= \sum_{i=1}^n x_i g_i(x) && \\

&= \sum_{i=1}^n \left[x_i\left(g_i(x) - \alpha_i\right)\right] + \sum_{i=1}^{n-1} \left[x_i \alpha_i\right] && \quad \text{ using } g_i(x) = \left(g_i(x) - \alpha_i\right) + \alpha_i \text{ and } \alpha_n = 0 \\

&= \left[\sum_{i=1}^n x_i\left(g_i(x) - \alpha_i\right)\right] + \left[\sum_{i=1}^{n-1} x_i x_n \alpha_i\right] + \left[\sum_{i=1}^{n-1} x_i \left(1 - x_n\right) \alpha_i\right] && \quad \text{ using } x_i = x_n x_i + x_i \left(1 - x_n\right). \\

\end{alignat}

Each of the $3\; n\; -\; 2$ terms above has the desired properties.

$\backslash blacksquare$

}}

• {{annotated link|Bump function}}
• {{annotated link|Continuously differentiable}}
• {{annotated link|Smoothness}}
• {{annotated link|Taylor's theorem}}

{{reflist|group=note}}

{{reflist}}

• {{cite book|author=Nestruev, Jet|title=Smooth manifolds and observables|publisher=Springer|location=Berlin|year=2002|isbn=0-387-95543-7}}
• {{Nestruev Smooth Manifolds and Observables 2020}}

Category:Real analysis

Category:Theorems in mathematical analysis