Exercise 3, § 3

Functions of a Real Variable, I, 40

With the notation of exerc. 2 and the same hypothesis on $[\mathbf{x}.y]$ suppose that each of the functions $g_k$ is $n$ times differentiable on I; for each function $\mathbf{f}$ which is $n$ times differentiable on I, with values in F, put

[g_0.\mathbf{f}] - [g_1.\mathbf{f}]' + [g_2.\mathbf{f}]'' + \cdots + (-1)^n [g_n.\mathbf{f}]^{(n)} = [h_0.\mathbf{f}] + [h_1.\mathbf{f}'] + \cdots + [h_n.\mathbf{f}^{(n)}],

which defines the functions $h_i$ ( $0 \leq i \leq n$ ) without ambiguity (exerc. 2); show that one has

[h_0.\mathbf{f}] - [h.\mathbf{f}]' + [h_2.\mathbf{f}]'' + \cdots + (-1)^n [h_n.\mathbf{f}]^{(n)} = [g_0.\mathbf{f}] + [g_1.\mathbf{f}'] + \cdots + [g_n.\mathbf{f}^{(n)}]

identically.

Solution

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