continuous linear mapping
If $({V}_{1},\parallel \cdot {\parallel}_{1})$ and $({V}_{2},\parallel \cdot {\parallel}_{2})$ are normed vector spaces^{}, a linear mapping $T:{V}_{1}\to {V}_{2}$ is continuous^{} if it is continuous in the metric induced by the norms.
If there is a nonnegative constant $c$ such that ${\parallel T(x)\parallel}_{2}\le c{\parallel x\parallel}_{1}$ for each $x\in {V}_{1}$, we say that $T$ is . This should not be confused with the usual terminology referring to a bounded function as one that has bounded^{} range. In fact, bounded linear mappings usually have unbounded^{} ranges.
The expression bounded linear mapping is often used in functional analysis^{} to refer to continuous linear mappings as well. This is because the two definitions are equivalent^{}:
If $T$ is bounded, then ${\parallel T(x)-T(y)\parallel}_{2}={\parallel T(x-y)\parallel}_{2}\le c{\parallel x-y\parallel}_{1}$, so $T$ is a Lipschitz function. Now suppose $T$ is continuous. Then there exists $r>0$ such that ${\parallel T(x)\parallel}_{2}\le 1$ when ${\parallel x\parallel}_{1}\le r$. For any $x\in {V}_{1}$, we then have
$$\frac{r}{{\parallel x\parallel}_{1}}{\parallel T(x)\parallel}_{2}={\parallel T\left(\frac{r}{{\parallel x\parallel}_{1}}x\right)\parallel}_{2}\le 1,$$ |
hence ${\parallel T(x)\parallel}_{2}\le r{\parallel x\parallel}_{1}$; so $T$ is bounded.
It can be shown that a linear mapping between two topological vector spaces^{} is continuous if and only if it is continuous at (http://planetmath.org/Continuous) $0$ [1].
References
- 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
Title | continuous linear mapping |
Canonical name | ContinuousLinearMapping |
Date of creation | 2013-03-22 13:15:41 |
Last modified on | 2013-03-22 13:15:41 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 7 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 46B99 |
Synonym | bounded linear mapping |
Related topic | HomomorphismsOfCAlgebrasAreContinuous |
Related topic | CAlgebra |
Related topic | BoundedLinearFunctionalsOnLpmu |
Defines | bounded linear transform |
Defines | bounded linear operator |