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I am learning the first rudiments of functional analysis and have a couple of queries.

1) A book says that the space W (1,p) is defined as a subset of functions u(x) of Lp (so far, good) such that the integral, for any smooth epsilon (x), of u(x) times the derivative of epsilon is equal to MINUS epsion times the derivative of u(x). I have to admit I do not get it.

2) Wikpedia reports the example of a non - continuous W (1,1) function on the unit ball in R3, namely u(x) = 1 / abs(x). Also it adds that "space Wk,p(Ω) will contain only continuous functions, but for which k this is already true depends both on p and on the dimension<...>Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball is "smaller" in higher dimensions". Why should the unti ball be "smaller" in higher dimensions?

Many thanks for your help

Muzialis

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