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I was given the statement in exam(for which I was supposed to mark it as correct or incorrect):

Every Finite dimensional space is reflexive. (I marked it as incorrect).

My attempt: I learnt the following theorem which says: Every finite dimensional normed space is reflexive. But since there was no use of word "normed", I thought it may happen that the definition of reflexivity may also be applicable to some other spaces where I could find a counterexample and so I marked it as incorrect.

My professor said, it is correct. So my question is : Is there any finite dimensional space which is not reflexive? Please throw some light.

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1 Answer 1

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Hints: (Prove them as easy exercises, if you haven't already)

  1. Any finite dimensional vector space can be given a norm.

  2. On a finite dimensional normed space, any two norms are equivalent.

Bingo!

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