examples of bounded and unbounded operators
The aim of this page is to list examples of bounded (http://planetmath.org/BoundedOperator) and unbounded linear operators^{}.
Bounded
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Shift operators on ${\mathrm{\ell}}^{p}$

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A linear operator is continuous^{} if and only if it is bounded (see this page (http://planetmath.org/ContinuousLinearMapping)).

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Any isometry is bounded.

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A multiplication operator $h(t)\mapsto f(t)h(t)$, where $f(t)$ is continuous and $h\in {L}^{p}[0,1]$.

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An integral operator $h(t)\mapsto {\int}_{0}^{1}K(t,s)h(s)\mathit{d}s$, where $$ and $h\in {L}^{2}[0,1]$. In fact this is a HilbertSchmidt operator.

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The Volterra operator $h(t)\mapsto {\int}_{0}^{t}h(s)\mathit{d}s$, where $h\in {L}^{p}[0,1]$.
Unbounded

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The derivative is an unbounded operator on the vector space^{} of smooth functions^{} equipped with the $\mathrm{sup}$norm.
Title  examples of bounded and unbounded operators 

Canonical name  ExamplesOfBoundedAndUnboundedOperators 
Date of creation  20130322 15:17:37 
Last modified on  20130322 15:17:37 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  12 
Author  matte (1858) 
Entry type  Example 
Classification  msc 47L25 