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continuous linear mapping
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(Definition)
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If $(V_1,\|\cdot\|_1)$ and $(V_2,\|\cdot\|_2)$ are normed vector spaces, a linear mapping $T:V_1\rightarrow V_2$ is continuous if it is continuous in the metric induced by the norms.
If there is a nonnegative constant $c$ such that $\|T(x)\|_2\leq c\|x\|_1$ for each $x\in V_1$ , we say that $T$ is bounded. 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 $\|T(x)-T(y)\|_2 = \|T(x-y)\|_2 \leq c\|x-y\|_1$ , 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 $0$ [1].
- 1
- W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
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"continuous linear mapping" is owned by Koro.
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Cross-references: topological vector spaces, Lipschitz function, equivalent, definitions, functional analysis, expression, unbounded, range, bounded, bounded function, metric induced by the norms, continuous, linear mapping, normed vector spaces
There are 24 references to this entry.
This is version 4 of continuous linear mapping, born on 2002-12-13, modified 2003-08-04.
Object id is 3741, canonical name is ContinuousLinearMapping.
Accessed 10806 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) |
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Pending Errata and Addenda
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