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As in the title I would like to clear my mind about what is meant by a basis in infinite dimension, especially in the case where the vector space is endowed with a compatible topology.
What I know is:

  • Hamel basis: a collection of linearly idependent vectors whose finite linear combinations express all the elements of the space. This notion should be independent of whether we place a topology on the vector space or not. We can always prove the existence of such a basis through Zorn's Lemma
  • Schauder basis: a collection of linearly independent vectors such that the closure of its span coincides with the whole space, in other terms we can express any vector of the space as an infinite sum. I also know this collection as a complete system. By its definition, it seems to me this concepts fits only in the case of Topological Vector Spaces. I do not know any existence results.
  • Orthonormal basis, or Fourier basis: a particular case of Schauder basis in separable Hilbert spaces. I know a characterization, which states that TFAE: B is a orthonormal basis, $B^\perp=0$, Parseval identity holds, the series of Fourier coefficients of any vector is $<\infty$ (aka Fischer-Riesz theorem).

What I would like to know is what are other concepts or generalization of the idea of basis in particular, if any, in the case of Topological Vector Spaces which are not normable, such as locally convex.

Thanks in advance

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  • $\begingroup$ Your notions are correct, but your question is way too broad. $\endgroup$ – uniquesolution Jul 16 '19 at 6:49
  • $\begingroup$ @uniquesolution ok, I restricted the scope of it $\endgroup$ – Francesco Bilotta Jul 16 '19 at 6:58
  • $\begingroup$ I think Schauder basis is good enough. Nevertheless, you forgot to require linear independency at each point. $\endgroup$ – Berci Jul 16 '19 at 8:21
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    $\begingroup$ Orthonormal bases make sense in nonseparable Hilbert spaces too: math.stackexchange.com/q/2827558/144766 $\endgroup$ – mechanodroid Jul 16 '19 at 9:42
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    $\begingroup$ In the book by Albiac and Kalton "Topics in Banach Space Theory" there is a chapter titled "Special Types of Bases". If you want further elaboration of the role of bases in topological vector spaces this may help you. $\endgroup$ – s.harp Jul 17 '19 at 11:39

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