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I was going through the definition of normed linear space on wikipedia. They have set the field to be $R$ and $C$ only. Even textbook I am using does so. Any specific purpose behind this. Thanks.

ogirkar
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    Possible duplicate of Normed vector spaces over finite fields – lulu Aug 05 '19 at 11:20
  • Ok it answered my question partially. I still have two doubts. – ogirkar Aug 05 '19 at 12:11
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    Please edit your post to include those doubts. Just to say, though: the "answer" to your question is that there are lots of properties of norms that work for the given cases but which fail over a general field. Sure there are analogs in other contexts and they are worth studying. But the general theory over $\mathbb R, \mathbb C$ is important to study on its own. – lulu Aug 05 '19 at 12:14
  • In the answer of linked question, it says if we consider finite field, valuation has to be trivial. Making field valuation trivial, will it give some absurd result. Means what's wrong if we set field valuation be 1 for all field elements.
    • – ogirkar Aug 05 '19 at 12:15
  • Well, it sure doesn't give an interesting topology. But of course not all norms are induced by a valuation (as an abstract matter). – lulu Aug 05 '19 at 12:16
  • Remember, the "point" of a norm is to give an interesting topology, one which respects the vector space structure in some helpful or interesting way. This is especially relevant for infinite dimensional vector spaces, for which there simply isn't an obvious (or unique) topology to consider. I scarcely need the theory of norms if I want to consider the discrete topology. – lulu Aug 05 '19 at 12:19
  • Ok thanks.second doubt was silly. Not mentioning. – ogirkar Aug 05 '19 at 12:20