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I know that it is possible to define a lot of different norms on continuous function spaces like $l_p$ norms. However, I was wondering if it is possible to define a norm on the function space consisting of all functions from $\Bbb R$ to $\Bbb R$. I have tried a few examples and none of them managed to fulfill all the requirements of being a norm, so maybe it is actually impossible to define such a norm.

Edit: I know such a norm exists but I wondered if it has a “simple” representation, meaning given a function I can actually calculate its norm and not just know “it exists”.

Hanul Jeon
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yotam maoz
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    There is a sensible topology for your space. The "product topology" or the "topology of pointwise convergence". But that topology is not induced by any norm. – GEdgar Mar 10 '21 at 11:22
  • It would be a pity if this question is closed before someone comes up with an answer providing a norm which has some relationship to the way the space is presented. In that sense this question is not exactly the same as the one pointed out by Jose. – Ruy Mar 10 '21 at 11:57
  • To be more precise, I'd be curious to know if there is a norm stronger than pointwise convergence. – Ruy Mar 10 '21 at 12:10

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