# Is a bounded operator necessarily linear?

The Wikipedia article for bounded operator is all about linear bounded operator. I was wondering

1. Can a bounded operator be non-linear? If yes, how is this defined?
2. Is a bounded operator generally assumed to be linear?

Thanks!

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2. Yes, many people use "operator" and "linear operator" to mean the same (it is tiresome to repeat "linear" every time). –  wildildildlife Apr 11 '11 at 19:07
Is Sine bounded? Is it linear? –  scineram Apr 12 '11 at 11:19

1. Yes, a bounded operator can be nonlinear. There are a lot of useful notions of `bounded non-linear operator'. One is that for an operator between topological spaces that the image of compact sets is compact. The operator $Tx = 1/(1-x)$ is bounded on $[0,\infty)$ under this definition, but so are a lot of nasty operators. It depends on what you are trying to get out of your operator.