# continuous linear mapping

If $T$ is bounded, then $\|T(x)-T(y)\|_{2}=\|T(x-y)\|_{2}\leq c\|x-y\|_{1}$, so $T$ is a Lipschitz function. Now suppose $T$ is continuous. Then there exists $r>0$ such that $\|T(x)\|_{2}\leq 1$ when $\|x\|_{1}\leq r$. For any $x\in V_{1}$, we then have

 $\frac{r}{\|x\|_{1}}\|T(x)\|_{2}=\|T\left(\frac{r}{\|x\|_{1}}x\right)\|_{2}\leq 1,$

hence $\|T(x)\|_{2}\leq r\|x\|_{1}$; 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) $0$ .

## 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