Derivative of a continuous linear map

Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$ and let $A \colon V \to W$ be a bounded linear map. Then $A$ is Fréchet differentiable at each point $x \in V$ , and its Fréchet derivative at $x$ is $A$ itself: $DA(x) = A$ .

Remarks

The proof is one line: $A(x + h) - A(x) - A h = 0$ for each $h$ , so the remainder is $0$ . For $V = \mathbb{R}^n$ and $W = \mathbb{R}^m$ , each linear map is bounded. So the derivative of $x \mapsto M x$ , for an $m \times n$ matrix $M$ , is the linear map with matrix $M$ at each point. An affine map $x \mapsto A x + b$ has the same derivative $A$ at each point, because the derivative of a constant map is $0$ .

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Euclidean space object
Fréchet derivative definition
Derivative of a continuous linear map statement

Tag	`0CYYAF8AD2`
URI	`m2://0CYYAF8AD2`
Kind	statement, proposition
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 0CYYAF8AD2`.
Uses	Fréchet derivative

Derivative of a continuous linear map

Statement

Remarks

Reading path

Further reading

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