Uniqueness of the derivative

Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$ , let $U \subseteq V$ be open, let $f \colon U \to W$ , and let $x \in U$ . If the bounded linear maps $A$ and $B$ from $V$ to $W$ are both Fréchet derivatives of $f$ at $x$ , then $A = B$ .

Remarks

The proof uses that $U$ is open. For a unit vector $v$ and small $t > 0$ , the point $x + t v$ lies in $U$ , and $\|t A v - t B v\| = o(t)$ . Divide by $t$ and let $t \to 0$ to get $A v = B v$ . So it is correct to speak of the derivative $Df(x)$ . On a set that is not open, for example a single point, a derivative within the set need not be unique.

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Tag	`0762WAE49P`
URI	`m2://0762WAE49P`
Kind	statement, proposition
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 0762WAE49P`.
Uses	Fréchet derivative

Uniqueness of the derivative

Statement

Remarks

Reading path

Further reading

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