Theorem of Banach-Steinhaus

The theorem of Banach-Steinhaus (also called Principle of the uniform terminal ) formed part, as well as the Theorem of Hahn-Banach and the Theorem of Banach-Schauder, of the fundamental results of the functional analysis. It was published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.

He affirms that for a family of continuous linear applications definite on a Espace of Banach is uniformly limited if and only if she is punctually limited. It is a very important consequence of the property of Baire, which spreads besides with the spaces of Fréchet.

The theorem

Statement

That is to say $E$ a Space of Banach and $F$ a vector Space normalized. One considers a family $(f_i) _ {I \ in I}$ of continuous linear applications of $E$ in $F$ . It is supposed that this family is punctually limited, i.e.:

$\ forall X \ in E, \ sup_ {I \ in I} ||f_i (X)|| < + \ infty$

Then $(f_i) _ {I \ in I}$ is uniformly limited, i.e. there exists a constant $K$ such as:

$\ forall I \ in I, ||f_i|| \ Leq K$

Demonstration

The proof rests on the fact that a Espace of Banach is a Espace of Baire, i.e. any countable meeting of closed of empty interior is of empty interior.

Let us consider $A_n$ the whole of the elements of $E$ such as $\ forall I \ in I, ||f_i (X)|| \ Leq n$ .

$A_n = \ bigcap_ {I \ in I} \ {X \ in E: ||f_i (X)|| \ Leq N \}$

$A_n$ is an intersection of closed, it one is thus closed. The family $(f_i) _ {I \ in I}$ is punctually limited, this assumption results in the equality ensemblist:

$E = \ bigcup_ {N \ in \ mathbb {NR}} A_n$

As $E$ is not empty interior, there exists $n_0 \ in \ mathbb {NR}$ such as $A_ {n_0}$ is not empty interior: it contains a ball of center $a$ and $R > 0$ .

Let us take a point $x$ $E$ located in the ball closed unit:

$\ forall I \ in I, ||f_i (X)|| = R ||f_i (\ frac {X} {R})|| \ Leq R ||f_i (a)|| + R ||f_i (+ \ frac {X} {R} has)|| \ Leq R (1 + n_0)$

i.e.: $(f_i) _ {I \ in I}$ is uniformly limited.

Alternative " forte" statement

Either $(T_ \ alpha) _ {\ alpha \ in has}$ a family of linear operators continuous of Banach in another. Then

or $\ sup_ {\ alpha \ in has} \ Vert T_ \ alpha \ Vert <+ \ infty$

or there exists residual U (i.e. a countable intersection the open dense ones; such a part is dense according to the theorem of Baire) such as

$\ forall X \ in U, \ sup_ {\ alpha \ in has} \ Vert T_ \ alpha (X) \ Vert=+ \ infty$

Examples of applications

Limit of a succession of continuous linear applications

Let us mention a very important corollary of the theorem of Banach-Steinhaus: if $(f_n)$ is a succession of continuous linear applications of the space of Banach $E$ in the vector space normalized $F$ which converges simply towards a function $f$ , then $f$ is also a continuous linear application.

Indeed, the linearity comes from a simple passage in extreme cases. And for all $x \ in E$ , $(f_n (X))$ converges, it is thus a limited continuation, and the theorem of Banach-Steinhaus affirms that $(f_n)$ is uniformly limited. $(f_n)$ is limited in standard subordinated by a constant $C$ , and by passage in extreme cases of the inequalities $f$ is limited of subordinate standard lower than $C$ .

Application to the sums of Riemann

See also: Somme of Riemann

That is to say E the space of the continuous functions on to actual values, provided with the standard $\| F \|_ \ infty = \ int_0^1 |F (T)| dt$ , and F = $\ mathbb R$ . For each entirety I , U _I the operator defined is by:

u_i (F) = I \ int_0^1 F (T) dt - \ sum_ {k=1} ^i F (k/i)

${u_i (F) \ over I}$ is not other than the error made in the calculation of the integral of F when one takes a sum of Riemann corresponding to a regular subdivision of in I equal intervals. This error is a $O ({1 \ over I})$ for the C¹ functions or lipschitziennes, but it is not the same for the continuous functions in general. Indeed, it is shown that $\| u_i \| = 2i$ , so that $\ sup_ {I \ in I} \| u_i \| = + \ infty$ and thus that the complementary one to has is dense. A function F pertaining to this complementary thus checks $\ sup_ {I \ in I} \|u_i (F) \| = + \ infty$ , which means that the unit $u_i (F)$ is not limited and thus that the made error ${u_i (F) \ over I}$ is not a $O ({1 \ over I})$ .
The theorem of Banach-Steinhaus gives a proof of the existence of objects checking such or such property, but this proof is not constructive.

Application to Fourier series

See also: Core of Dirichlet

If $f$ is a function (let us say continuous) period $2 \ pi \,$ , it is checked that the sum partial $n$ -ième of its Fourier series is
$S_n (F) (X) = \ frac {1} {2 \ pi} \ int_ {- \ pi} ^ \ pi F (T) D_n (x-t) dt$ , with $D_n (T) = \ displaystyle {\ frac {\ sin (2n+1) \ frac {T} {2}} {\ sin \ frac {T} {2}}}$ (Core of Dirichlet)
For $n$ fixed, the standard of the application $f \ mapsto S_n (F) (X)$ , sight like forms linear on the space of the continuous functions and period $2 \ pi$ , provided standard sup, is equal to $\ frac {1} {2 \ pi} \ int_ {- \ pi} ^ \ pi \ green D_n (T) dt$
It is checked that the number $L_n= \ int_ {- \ pi} ^ \ pi \ green D_n (T) \ green dt$ called constant of Lebesgue, towards the infinite one like $\ log (N)$ tends.
According to the theorem of Banach-Steinhaus, there thus exists a function $f \,$ such as $\ green S_n (F) (X) \ green$ tends towards the infinite one when $n \,$ tends towards the infinite one. Thus, the Fourier series of $f \,$ diverge in $x \,$ .
If one uses the strong version of the theorem of Banach-Steinhaus, one even sees that the whole of the continuous functions of period $2 \ pi \,$ of which the Fourier series diverge in $x \,$ is dense for the topology of uniform convergence.
This argument is all the more remarkable as it is not very easy to find explicit examples.

See too

Stefan Banach, Hugo Steinhaus, " On the principle of the condensation of singularités". Fundamenta Mathematicae

Applications of the Theorem of Banach-Steinhaus on limps in Baire.

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