Нүктелік шектеулілік біркелкі шектеулілікке келіп тиімді теорема

A theorem stating that pointwise boundedness implies uniform boundedness

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.

The completeness of enables the following short proof, using the Baire category theorem. There are also simple proofs not using the Baire theorem .

In a topological vector space (TVS) "bounded subset" refers specifically to the notion of a von Neumann bounded subset. If happens to also be a normed or seminormed space, say with (semi)norm then a subset is (von Neumann) bounded if and only if it is norm bounded, which by definition means

Although the notion of a nonmeager set is used in the following version of the uniform bounded principle, the domain is not assumed to be a Baire space. Every proper vector subspace of a TVS has an empty interior in So in particular, every proper vector subspace that is closed is nowhere dense in and thus of the first category (meager) in (and the same is thus also true of all its subsets). Consequently, any vector subspace of a TVS that is of the second category (nonmeager) in must be a dense subset of (since otherwise its closure in would a closed proper vector subspace of and thus of the first category).

proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces.