Theorem of Banach-Schauder

In analyzes functional, the theorem of Banach-Schauder , also called theorem of the open application is a fundamental result which affirms that a Linear application continues surjective between two normalized vector spaces Complet S is open. It is an important consequence of the theorem of Baire, which affirms that in a complete metric space (and thus in particular in a Espace of Banach), any countable intersection of open dense is dense, which makes it possible to generalize the theorem of Banach-Schauder to the spaces of Fréchet.

Statement

That is to say $E$ and $F$ two spaces of Banach and $f$ a linear application continue $E$ towards $F$.

If $f$ is surjective, then $f$ is open, i.e the image of all open $E$ by $f$ is open of $F$.

Demonstration

To show that $f$ is opened, it is enough by linearity to show that the image of any vicinity of $0$ (in $E$) by $f$ is a vicinity of $0$ (in $F$), i.e

$\backslash \; forall\; \backslash \; varepsilon\; >\; 0,\; \backslash \; exists\; \backslash \; eta\; >\; 0,\; B\_F\; (0,\; \backslash \; eta)\; \backslash \; subset\; F\; (B\_E\; (0,\; \backslash \; varepsilon))$

(By homogeneity of $f$, it is even enough to make it for only one $\backslash \; varepsilon$). One introduces the closed following ones:

$F\_n\; =\; \backslash \; overline\; \{F\; (B\_E\; (0,\; N))\}$

As $f$ is surjective, one has the equality ensemblist:

$F\; =\; \backslash \; bigcup\_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}\; F\_n$

$F$ is a space of Banach, in particular it checks the property of Baire, therefore one of these closed, $F\_N$ is of nonempty interior: it contains a ball $B\_F\; (there,\; \backslash \; eta)$.

Closed the $F\_\; \{2N\}$ thus contains the ball $B\_F\; (0,\; \backslash \; eta)$. By homogeneity of $f$ one has an entirety $M$ thus such as:

$B\_F\; (0,1)\; \backslash \; subset\; \backslash \; overline\; \{F\; (B\_E\; (0,\; M))\}$

It does not remain any more that to make jump the bar. By homogeneity of $f$, one deduces from this result that:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; B\_F\; (0,1/2^n)\; \backslash \; subset\; \backslash \; overline\; \{F\; (B\_E\; (0,\; M/2^n))\}$

Let us show that $B\_F\; (0,1)\; \backslash \; subset\; F\; (B\_E\; (0,3M))$. For that, we give a $z\; \backslash \; in\; B\_F\; (0,1)$

• It exists $x\_0$ of standard lower than $M$ such as $z\_1\; =\; Z\; -\; F\; (x\_0)$ is of standard lower than 1/2.
• There exists $x\_1$ of standard lower than $M/2$ such as $z\_2\; =\; z\_1\; -\; F\; (x\_1)$ is of standard lower than 1/4.

One builds by recurrence a continuation $(x\_n)$ of points of $E$ such as $||x\_n||\; \backslash \; Leq\; M/2^n$ and $z\_n\; =\; Z\; -\; F\; (x\_0\; +\; \backslash \; cdots\; +\; x\_n)$ is of standard lower than $1/2^n$.

The series $\backslash \; sum\; x\_n$ is absolutely convergent, therefore as $E$ is a space of Banach, it converges. Moreover,

$\backslash Big|\backslash \; Big|\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; x\_n\; \backslash \; Big|\backslash \; Big|\; \backslash \; Leq\; \backslash \; sum\_\; \{N\; =\; 0\}\; ^\; \{+\; \backslash \; infty\}\; ||x\_n||\; \backslash \; Leq\; M\; \backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{1\}\; \{2^n\}\; =\; 2M$

And, by passage in extreme cases:

$z\; =\; F\; \backslash \; Big\; (\backslash \; sum\_\; \{n=0\}\; ^\; \{+\; \backslash \; infty\}\; x\_n\; \backslash \; Big)\; \backslash \; in\; F\; (\backslash \; overline\; \{B\_E\; (0,2M)\})\; \backslash \; subset\; F\; (B\_E\; (0,3M))$

It is what it was necessary to show.

Consequences

Theorem of isomorphism of Banach

The theorem of Banach-Schauder has a fundamental consequence (in fact, it is of an equivalent form of the theorem, and not about a result weaker), called theorem of isomorphism of Banach , theorem of Baire-Banach or more simply theorem of Banach :

If $f$ is a continuous bijective linear application between two spaces of Banach, then $f$ is a Homéomorphisme.

Theorem of the closed graph

See also: Theorem of the graph closed

The theorem of Banach-Schauder is also at the origin of a powerful criterion of continuity of the linear applications between two spaces of Banach, it acts of the theorem of the closed graph:

Is $E$ and $F$ two spaces of Banach, and $f$ a linear application of $E$ in $F$. $f$ is continuous if and only if its graph is a closed part of $E\; \backslash \; times\; F$.

Example of application

Are $E=L^1\; (S^1)\; \backslash ,$ the space of Banach of the integrable functions on the circle, and $F=c\_0\; (\backslash \; mathbb\; \{Z\})\; \backslash ,$ the space of the complex continuations indexed by entireties relative and tending towards zero. The application $f\; \backslash \; mapsto\; \backslash \; big\; (\backslash \; widehat\; \{F\}\; (N)\; \backslash \; big)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\}$ which associates with the function F the continuation of its coefficients of Fourier is continuous and injective of E in F , but is not surjective. Indeed, if such were the case, there would exist a constant $C>O\; \backslash ,$ such as, for any function $f\; \backslash \; in\; E$, $\backslash \; Green\; F\; \backslash \; Vert\_1\; \backslash \; C\; \backslash \; sup\_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\}\; \backslash \; green\; \backslash \; widehat\; \{F\}\; (N)\; \backslash \; vert$ By applying such an inequality following the cores of Dirichlet $D\_k\; \backslash ,$, one arrives at a contraction. Indeed, $\backslash \; Green\; D\_k\; \backslash \; Vert\_1=0\; (\backslash \; log\; (K))$ whereas them $\backslash \; green\; \backslash \; widehat\; \{D\_k\}\; (N)\; \backslash \; vert$ is limited by 1 .

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