Hadamard's lemma

In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner.

Statement

Hadamard's lemma — Let [math]\displaystyle{ f }[/math] be a smooth, real-valued function defined on an open, star-convex neighborhood [math]\displaystyle{ U }[/math] of a point [math]\displaystyle{ a }[/math] in [math]\displaystyle{ n }[/math]-dimensional Euclidean space. Then [math]\displaystyle{ f(x) }[/math] can be expressed, for all [math]\displaystyle{ x \in U, }[/math] in the form: [math]\displaystyle{ f(x) = f(a) + \sum_{i=1}^n \left(x_i - a_i\right) g_i(x), }[/math] where each [math]\displaystyle{ g_i }[/math] is a smooth function on [math]\displaystyle{ U, }[/math] [math]\displaystyle{ a = \left(a_1, \ldots, a_n\right), }[/math] and [math]\displaystyle{ x = \left(x_1, \ldots, x_n\right). }[/math]

Proof

Proof

Let [math]\displaystyle{ x \in U. }[/math] Define [math]\displaystyle{ h : [0, 1] \to \R }[/math] by [math]\displaystyle{ h(t) = f(a + t(x - a)) \qquad \text{ for all } t \in [0, 1]. }[/math]

Then [math]\displaystyle{ h'(t) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(a + t(x - a)) \left(x_i - a_i\right), }[/math] which implies [math]\displaystyle{ \begin{aligned}h(1) - h(0)&= \int_0^1 h'(t)\,dt\\ &= \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} }[/math]

But additionally, [math]\displaystyle{ h(1) - h(0) = f(x) - f(a), }[/math] so by letting [math]\displaystyle{ g_i(x) = \int_0^1 \frac{\partial f}{\partial x_i}(a + t(x - a))\, dt, }[/math] the theorem has been proven. [math]\displaystyle{ \blacksquare }[/math]

Consequences and applications

Corollary — If [math]\displaystyle{ f : \R \to \R }[/math] is smooth and [math]\displaystyle{ f(0) = 0 }[/math] then [math]\displaystyle{ f(x)/x }[/math] is a smooth function on [math]\displaystyle{ \R. }[/math] Explicitly, this conclusion means that the function [math]\displaystyle{ \R \to \R }[/math] that sends [math]\displaystyle{ x \in \R }[/math] to [math]\displaystyle{ \begin{cases} f(x)/x & \text{ if } x \neq 0 \\ \lim_{t \to 0} f(t)/t & \text{ if } x = 0 \\ \end{cases} }[/math] is a well-defined smooth function on [math]\displaystyle{ \R. }[/math]

Proof

By Hadamard's lemma, there exists some [math]\displaystyle{ g \in C^{\infty}(\R) }[/math] such that [math]\displaystyle{ f(x) = f(0) + x g(x) }[/math] so that [math]\displaystyle{ f(0) = 0 }[/math] implies [math]\displaystyle{ f(x)/x = g(x). }[/math] [math]\displaystyle{ \blacksquare }[/math]

Corollary — If [math]\displaystyle{ y, z \in \R^n }[/math] are distinct points and [math]\displaystyle{ f : \R^n \to \R }[/math] is a smooth function that satisfies [math]\displaystyle{ f(z) = 0 = f(y) }[/math] then there exist smooth functions [math]\displaystyle{ g_i, h_i \in C^{\infty}\left(\R^n\right) }[/math] ([math]\displaystyle{ i = 1, \ldots, 3n - 2 }[/math]) satisfying [math]\displaystyle{ g_i(z) = 0 = h_i(y) }[/math] for every [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ f = \sum_{i}^{} g_i h_i. }[/math]

Proof

By applying an invertible affine linear change in coordinates, it may be assumed without loss of generality that [math]\displaystyle{ z = (0, \ldots, 0) }[/math] and [math]\displaystyle{ y = (0, \ldots, 0, 1). }[/math] By Hadamard's lemma, there exist [math]\displaystyle{ g_1, \ldots, g_n \in C^{\infty}\left(\R^n\right) }[/math] such that [math]\displaystyle{ f(x) = \sum_{i=1}^n x_i g_i(x). }[/math] For every [math]\displaystyle{ i = 1, \ldots, n, }[/math] let [math]\displaystyle{ \alpha_i := g_i(y) }[/math] where [math]\displaystyle{ 0 = f(y) = \sum_{i=1}^n y_i g_i(y) = g_n(y) }[/math] implies [math]\displaystyle{ \alpha_n = 0. }[/math] Then for any [math]\displaystyle{ x = \left(x_1, \ldots, x_n\right) \in \R^n, }[/math] [math]\displaystyle{ \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} }[/math] Each of the [math]\displaystyle{ 3 n - 2 }[/math] terms above has the desired properties. [math]\displaystyle{ \blacksquare }[/math]

See also

• Bump function – Smooth and compactly supported function
• Smoothness – Number of derivatives of a function (mathematics)
• Taylor's theorem – Approximation of a function by a truncated power series

References

• Nestruev, Jet (2002). Smooth manifolds and observables. Berlin: Springer. ISBN 0-387-95543-7. 
• Template:Nestruev Smooth Manifolds and Observables 2020