manifolds and cobordisms
cobordism theory, Introduction
Definitions
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The Fréchet derivative is a kind of functional derivative.
There are various ways to set up differentiation in finite dimensions, the most common being the total derivative and the directional derivatives. In infinite dimensions, these become the Fréchet derivative and the Gâteaux derivative respectively.
The definition of the Fréchet derivative of a function is a generalisation of the notion of the total derivative of a function in finite dimensions. In finite dimensions, the total derivative of a function at a point is defined to be (assuming that it exists) the unique linear operator such that:
As all norms in finite dimensions are norm-equivalent, the choice of norm does not matter.
This generalises most easily to normed vector spaces. As it involves limits, it is generally most convenient to work with Banach spaces.
Let and be Banach spaces with norms and respectively. Let be an open subset. A (possibly non-linear) function is said to be Fréchet differentiable at if there is a continuous linear operator such that:
The operator is called the Fréchet derivative of at and is written or .
If the Fréchet derivative exists, it is unique.
With the notation of Definition , a function is said to be of class on if it is Fréchet differentiable throughout and the resulting function is continuous as a function , where is the Banach space of continuous linear operators from to .
The differentiability of can then be questioned, and for , the class is defined by iteration in the obvious way.
Last revised on November 29, 2017 at 13:38:45. See the history of this page for a list of all contributions to it.