Are any two infinite-dim. V.Spaces isomorphic?

disregardthat

Quote by Landau

and this is indeed "obviously" countable as Q,B and N are. To prove this, I would say given n, there are Q^n B^n linear combinations of n basis elements.

Q^nB^n is an upper bound at least, counting them in this way does not give an injection. But of course, as you know the space is at least countable, we are done.

Landau

True. But you probably mean "at most countable" instead of "at least countable".

disregardthat

Quote by Landau

True. But you probably mean "at most countable" instead of "at least countable".

I meant at least countable (countable basis). My point was that you gave an argument for why it is at most countable, so we conclude it is countable.

Landau

Sorry, I didn't read carfully. I agree.

russel

Hello there, I'd like to ask if the Φ space (the one where each element is a sequence of finite non-zero terms) with norm 1 is isomorphic to Φ space with norm 2. Is it or not? And why? Has this to do with the fact that Φ is never Banach?

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Landau Recognitions: Science Advisor	True. But you probably mean "at most countable" instead of "at least countable".

russel	Hello there, I'd like to ask if the Φ space (the one where each element is a sequence of finite non-zero terms) with norm 1 is isomorphic to Φ space with norm 2. Is it or not? And why? Has this to do with the fact that Φ is never Banach?