# Uniform boundedness principle

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Uniform boundedness principle

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.

## Uniform boundedness principle

The precise statement of the result is:

Theorem. Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have

$\sup_{T \in F} \|T (x)\| < \infty,$

then

$\sup_{T \in F} \|T\| < \infty.$

The completeness of X enables the following short proof, using the Baire category theorem:

Proof.

Define the closed sets Xn with n = 1, 2, 3, … by
$X_n = \{ x \in X : \sup_{T \in F} \|T(x)\| \le n \}.$
By hypothesis, the union of all the Xn is X. Since X is a Baire space, one of the Xn, say Xm, has an interior point (Xn are closed sets), i.e., there exists a δ > 0 and a y in X such that all x ∈ X with ||xy|| < δ are elements of Xm. Now, choose an arbitrary z in X with ||z|| < δ. Then, y and y + z are elements of Xm and hence, for every operator T in the family F, ||T(z)|| ≤ ||T(y + z)|| + ||T(y)|| ≤ m + m = 2m. Since z is arbitrary in the ball of radius δ, it follows that ||T|| ≤ 2m / δ for all T in F, which proves the theorem.

A direct consequence is:

Corollary. If a sequence of bounded operators (Tn) converges pointwise, that is, lim Tn(x) exists for all x in X, then these pointwise limits define a bounded operator T.

Note it is not claimed above that Tn converges to T in operator norm, i.e. uniformly on bounded sets. (However, since {Tn} is bounded in operator norm, and the limit operator T is continuous, a standard Hahn–Banach theorem.

Let L(XY) denote the continuous operators from X to Y, with the operator norm. If the collection F is unbounded in L(XY), then by the uniform boundedness principle, the set

$R = \{ x \in X : \sup_{T \in F} \|T(x)\| = \infty \}$

is not empty. In fact, it is dense in X. The complement of R in X is the countable union of closed sets ∪Xn. By the argument used in proving the theorem, each Xn is nowhere dense, i.e. the subset ∪Xn is of first category. Therefore R is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows:

Theorem. Let X be a Banach space, {Yn} a sequence of normed vector spaces, and Fn a unbounded family in L(X, Yn). Then the set

$R = \{ x \in X : \sup_{T \in F_n} \|T (x)\| = \infty, \; \mbox{for all} \; n \}$

is dense in X.

Proof.

The complement of R is the countable union
$\cup_{n,m} \{ x \in X : \sup_{T \in F_n} \|T(x)\| \le m \}$
of sets of first category. Therefore its residual set R is dense.

### An example: pointwise convergence of Fourier series

Let T be the circle, and let C(T) be the Banach space of continuous functions on T, with the uniform norm. Using the uniform boundedness principle, one can show that the Fourier series, "typically", does not converge pointwise for elements in C(T).

For ƒ in C(T), its Fourier series is defined by

$\sum_{k \in \mathbb{Z}} \hat{f}(k) \mathrm{e}^{ikx} = \sum_{k \in \mathbb{Z}}\frac{1}{2 \pi} \Bigl( \int_0 ^{2 \pi} f(t) \mathrm{e}^{-ikt} dt \Bigr) \mathrm{e}^{ikx},$

and the N-th symmetric partial sum is

$S_N(f)(x) = \sum_{ -N \leq k \leq N } \hat{f}(k) \mathrm{e}^{ikx} = \frac{1}{2 \pi} \int_0 ^{2 \pi} f(t) D_N(x - t) \, dt,$

where DN is the N-th Dirichlet kernel. Fix x in T and consider the convergence of {SN(ƒ)(x)}. The functional φN, x : C(T) → C defined by

$\varphi_{N, x} (f) = S_N(f)(x), \quad f \in C(\mathbf{T}),$

is bounded. The norm of φN, x, in the dual of C(T), is the norm of the signed measure (2π)−1DN(xt) dt, namely

$\| \varphi_{N, x} \| = \frac{1}{2 \pi} \int_0 ^{2 \pi} | D_N(x-t) | \, dt = \frac{1}{2 \pi} \int_0 ^{2 \pi} | D_N(s) | \, ds = \| D_N \|_{L^1(\mathbf{T})}.$

One can verify that

$\frac{1}{2 \pi} \int_0 ^{2 \pi} | D_N(t) | \, dt \ge \int_0^\pi \frac{\bigl|\sin\bigl( (N+1/2)t \bigr)\bigr|} t \, dt \rightarrow \infty.$

So the collection {φN, x} is unbounded in C(T), the dual of C(T). Therefore by the uniform boundedness principle, for any x in T, the set of continuous functions whose Fourier series diverges at x is dense in C(T).

More can be concluded by applying the principle of condensation of singularities. Let {xm} be a dense sequence in T. Define

$\varphi_{N, x_m}$

in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each xm is dense in C(T) (however, the Fourier series of a continuous function ƒ converges to ƒ(x) for almost every x ∈ T, by Carleson's theorem).

## Generalizations

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):

• Given a barrelled space X and a locally convex space Y, then any family of pointwise bounded continuous linear mappings from X to Y is equicontinuous (even uniformly equicontinuous).

Alternatively, the statement also holds whenever X is a Baire space and Y is a locally convex space (Shtern 2001).

Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically,

• Let X be a Fréchet space, Y a normed space, and H a set of continuous linear mappings of X into Y. If $\sup_{u\in H}\|u(x)\|<\infty$ for every xX, then the family H is equicontinuous.

## References

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