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].

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