Uniform boundedness principle |
In mathematics, the uniform boundedness principle (sometimes
known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. In its basic form, it asserts that for a family of continuous linear
operators whose domain is a Banach space, pointwise boundedness is equivalent to boundedness.
More precisely, let X be a Banach
space and N be a normed
vector space. Suppose that F is a collection of continuous linear operators from
X to N. The uniform boundedness principle states that if
for all x in X we have
-
then
-
In some texts, one finds this called the Banach-Steinhaus Theorem, since it is a generalisation of a theorem first
appearing in a 1927 paper of Stefan
Banach and Hugo Dyonizy Steinhaus. The uniform
boundedness principle is often considered one of the three cornerstone theorems of functional analysis, the others being the
Hahn-Banach theorem and the open mapping theorem.
Using the Baire category theorem, we have the
following short proof:
- For n = 1,2,3, ... let Xn = { x : ||T(x)|| ≤ n
(∀ T ∈ F) } . By hypothesis, the union of all the Xn is X.
- Since X is a Baire space, one of the Xn has
an interior point, giving some δ > 0 such that ||x|| < δ ⇒ x ∈
Xn.
- Hence for all T ∈ F, ||T|| < n/δ, so that n/δ is a uniform
bound for the set F.
A version of the uniform boundedness principle also holds for F-spaces, with
uniform boundedness being replaced by uniform
equicontinuity.
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