Derivative of a map into a product

Statement

Let $V$ , $W_1$ and $W_2$ be normed vector spaces over $\mathbb{R}$ , let $U \subseteq V$ be open, and let $x \in U$ . Let $f_1 \colon U \to W_1$ and $f_2 \colon U \to W_2$ be Fréchet differentiable at $x$ . Then the map $F \colon U \to W_1 \times W_2$ , $F(y) = (f_1(y), f_2(y))$ , is Fréchet differentiable at $x$ , and $DF(x)(h) = \big(Df_1(x)(h), \, Df_2(x)(h)\big) \quad \text{for each } h \in V.$ Here $W_1 \times W_2$ has the norm $\|(w_1, w_2)\| = \max(\|w_1\|, \|w_2\|)$ .

Remarks

The converse also holds: if $F$ is differentiable at $x$ , then so are its two components, because each projection $W_1 \times W_2 \to W_i$ is a bounded linear map. Any norm on $W_1 \times W_2$ that is equivalent to the max norm gives the same derivative. Applied $m$ times with $W_i = \mathbb{R}$ , this gives the derivative of a map into R^n by components.

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Euclidean space object
Fréchet derivative definition
Derivative of a map into a product statement

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Derivative of a map into R^n by components

Tag	`0QQMXFTX0A`
URI	`m2://0QQMXFTX0A`
Kind	statement, proposition
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 0QQMXFTX0A`.
Uses	Fréchet derivative

Derivative of a map into a product

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Remarks

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