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Sosha
Sosha
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I havewant to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper by NAM-KIu TSING solving the same problem for Banach space. But, the proof is not much intuitive. Is it possible to give a easier proof for Hilbert space ?

Using proof by contradiction, the aim is to somehow find a Cauchy sequence and then use completeness to get a limit and show that cauchy sequence does not converge to that limit.

Thanks

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper by NAM-KIu TSING solving the same problem for Banach space. But, the proof is not much intuitive. Is it possible to give a easier proof for Hilbert space ?

Using proof by contradiction, the aim is to somehow find a Cauchy sequence and then use completeness to get a limit and show that cauchy sequence does not converge to that limit.

Thanks

I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper by NAM-KIu TSING solving the same problem for Banach space. But, the proof is not much intuitive. Is it possible to give a easier proof for Hilbert space ?

Using proof by contradiction, the aim is to somehow find a Cauchy sequence and then use completeness to get a limit and show that cauchy sequence does not converge to that limit.

Thanks

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Sosha
Sosha
  • 317
  • 2
  • 6

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper by NAM-KIu TSING solving the same problem for Banach space. But, the proof is not much intuitive. Is it possible to give a easier proof for Hilbert space ?

Using proof by contradiction, the aim is to somehow find a Cauchy sequence and then use completeness to get a limit and show that cauchy sequence does not converge to that limit.

Thanks