quality:correction, stylistic:style_transfer, information:classification
Well-definedness as a distribution

To prove the existence of the limit

for a Schwartz function u (x) {\displaystyle u(x)}, first observe that u (x) − u (− x) x {\displaystyle {\frac {u(x)-u(-x)}{x}}} is continuous on [ 0, ∞), {\displaystyle [0,\infty),} as

and hence

since u ′ (x) {\displaystyle u'(x)} is continuous and L'Hopital's rule applies.

Therefore, ∫ 0 1 u (x) − u (− x) x d x {\displaystyle \int _{0}^{1}\,{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x} exists and by applying the mean value theorem to u (x) − u (− x), {\displaystyle u(x)-u(-x),} we get:

And furthermore:

we note that the map

is bounded by the usual seminorms for Schwartz functions u {\displaystyle u}. Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.

Note that the proof needs u {\displaystyle u} merely to be continuously differentiable in a neighbourhood of 0 and x u {\displaystyle x\,u} to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as u {\displaystyle u} integrable with compact support and differentiable at 0.