Limited operator

In mathematics , linear mappings between normalized vector spaces are called bounded (linear) operators if their operator norm is finite. Linear operators are bounded if and only if they are continuous , which is why bounded linear operators are often called continuous (linear) operators .

Definitions

Let and be standardized vector spaces . A linear operator is a linear mapping . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T\colon X\to Y}$

A bounded operator is a linear operator for which there is one with for all . ${\displaystyle T\colon X\to Y}$ ${\displaystyle M}$ ${\displaystyle \Vert Tx\Vert \leq M\Vert x\Vert }$ ${\displaystyle x\in X}$

The smallest constant with for all is called the norm of . For them applies ${\displaystyle M}$ ${\displaystyle \Vert Tx\Vert \leq M\Vert x\Vert }$ ${\displaystyle x\in X}$ ${\displaystyle \Vert T\Vert }$ ${\displaystyle T}$

${\displaystyle \Vert T\Vert =\sup _{\Vert x\Vert =1}\Vert Tx\Vert }$

and for all the inequality ${\displaystyle x\in X}$

${\displaystyle \Vert Tx\Vert \leq \Vert T\Vert \Vert x\Vert }$.

continuity

A linear operator is restricted if and only if it is continuous, i.e. if it fulfills one of the following equivalent conditions:

• if so, in the metric induced by the respective norm , ${\displaystyle x_{n}\to x}$ ${\displaystyle Tx_{n}\to Tx}$
• for everyone and everyone there is one with ${\displaystyle x_{0}\in X}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \delta >0}$

${\displaystyle \Vert x-x_{0}\Vert <\delta \Rightarrow \Vert Tx-Tx_{0}\Vert <\epsilon }$,

• Archetypes of open sets are open.

Bounded linear operators are therefore often referred to as continuous linear operators. If linearity is assumed, one often speaks only of continuous operators or bounded operators. If the image space is the scalar field, one says functional instead of operator.

Furthermore, the following statements are equivalent:

• ${\displaystyle T}$ is steady.
• ${\displaystyle T}$is continuous in 0 .
• ${\displaystyle T}$is uniformly continuous .
• ${\displaystyle T}$ is limited.

Examples

• If is finite-dimensional , then every linear operator is continuous. ${\displaystyle X}$ ${\displaystyle T\colon X\to Y}$
• If one has two norms on the same vector space, then the norms are equivalent if and only if the identity maps are continuous in both directions.
• The functional defined by is continuous with , whereby, as usual , is provided with the supreme norm . ${\displaystyle T(f):=f(0)}$ ${\displaystyle T\colon C(\left[0,1\right],\mathbb {R} )\to \mathbb {R} }$ ${\displaystyle \Vert T\Vert =1}$ ${\displaystyle C(\left[0,1\right],\mathbb {R} )}$
• The functional defined by is continuously with . ${\displaystyle T(f):=f(0)+f^{\prime }(0)}$ ${\displaystyle T\colon C^{1}(\left[0,1\right],\mathbb {R} )\to \mathbb {R} }$ ${\displaystyle \Vert T\Vert =1}$
• The functional defined by is continuously with . ${\displaystyle \textstyle T(f):=\int _{0}^{1}f(x)dx}$ ${\displaystyle T\colon C(\left[0,1\right],\mathbb {R} )\to \mathbb {R} }$ ${\displaystyle \Vert T\Vert =1}$
• From the Hölder inequality it follows that for the functional defined by is continuous with . ${\displaystyle g\in L^{q}(\mathbb {R} )}$ ${\displaystyle \textstyle T(f):=\int _{\mathbb {R} }fg}$ ${\displaystyle T\colon L^{p}(\mathbb {R} )\to \mathbb {R} }$ ${\displaystyle \Vert T\Vert =\Vert g\Vert _{L^{q}}}$
• The integral operator defined by a continuous function and is continuous and the inequality applies . ${\displaystyle k\colon \left[0,1\right]^{2}\to \mathbb {R} }$ ${\displaystyle \textstyle Tf(x):=\int _{0}^{1}k(x,y)f(y)da}$ ${\displaystyle T\colon C(\left[0,1\right])\to C(\left[0,1\right])}$ ${\displaystyle \Vert T\Vert \leq \Vert k\Vert _{\infty }}$
• The differential operator on is not a continuous operator for the supremum norm. For example is , but . But the operator is continuous as an operator . ${\displaystyle {\tfrac {d}{dx}}}$ ${\displaystyle C^{1}\left[0,1\right]}$ ${\displaystyle \Vert x^{n}\Vert _{\infty }=1}$ ${\displaystyle \Vert {\tfrac {d}{dx}}x^{n}\Vert _{\infty }=n}$ ${\displaystyle {\tfrac {d}{dx}}\colon C^{1}(\left[0,1\right])\to C(\left[0,1\right])}$

The space of continuous operators

Let be normalized vector spaces. Then ${\displaystyle X,Y}$

${\displaystyle L(X,Y)=\left\{T\colon X\to Y\mid T{\mbox{ ist linear und stetig}}\right\}}$

with the operator norm a normalized vector space. ${\displaystyle \Vert .\Vert }$

If is complete , then is complete. ${\displaystyle Y}$ ${\displaystyle L(X,Y)}$

If a dense subspace and is complete, then each continuous operator has a unique continuous extension with . ${\displaystyle D\subset X}$ ${\displaystyle Y}$ ${\displaystyle T\in L(D,Y)}$ ${\displaystyle {\widehat {T}}\in L(X,Y)}$ ${\displaystyle \Vert {\widehat {T}}\Vert =\Vert T\Vert }$

Constrained linear operators between topological vector spaces

Analogous to the above definition, one calls a linear operator between topological vector spaces and bounded, if the image of each bounded subset is bounded. ${\displaystyle T\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

If and are additionally locally convex vector spaces , then the bounded operator is continuous, precisely because if is a bornological space . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T\colon X\to Y}$ ${\displaystyle X}$

Bounded mappings between topological vector spaces

→ Main article : Restricted figure

In some cases, non-linear mappings between vector spaces are also referred to as (non-linear) operators in German literature.

So if and are topological vector spaces , then a mapping is called bounded if the image of every bounded subset is bounded. ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle T\colon V\to W}$

literature

• Harro Heuser : Functional Analysis. Theory and application. Teubner, Wiesbaden 1975, 4th revised edition, ibid 2006, ISBN 3-8351-0026-2 .
• Dirk Werner : Functional Analysis. Springer Verlag, ISBN 3642210163 .

Individual evidence

1. ^ Norbert Adasch, Bruno Ernst, Dieter Keim: Topological Vector Spaces: The Theory Without Convexity Conditions . Springer-Verlag, Berlin Heidelberg, ISBN 978-3-540-08662-8 , pp. 60 .
2. ^ Klaus Deimling: Nonlinear equations and degrees of mapping. Springer-Verlag, Berlin / Heidelberg / New York 1974, ISBN 3-540-06888-0 .