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Let $X$ be reflexive Banach space or if you need one could assume that $X=H$ where $H$ is a usual separable Hilbert space and $\{e_n\}_{n = 1}^\infty$ be a Schauder basis in it which means for any x $\in X$ you have a unique representation $x = \sum_{n = 1} ^{\infty} x_ne_n$.

By saying that system $\{f_n\}_{n=1}^{\infty}$ is adjoint to $\{e_n\}_{n = 1}^{\infty}$ I mean a biorthogonal system associated with $\{e_n\}_{n=1}^{\infty}$ i.e. $(f_i, e_j) = \delta_{ij}$ for $\forall$ i, j.

My question is would adjoint system $\{f_n\}_{n=1}^{\infty}$ associated to arbitrary Schauder basis $\{e_n\}_{n=1}^{\infty}$ be Schauder basis in $X^*$ (or in $H$ respectively) itself or not?

I'll just add that answer is positive if you'd ask the same question for complete, minimal system or Riesz basis.

Thank you for any advices and remarks.

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  • $\begingroup$ In the Hilbert space case, it follows $f_i=e_i$. $\endgroup$
    – daw
    Commented Feb 8, 2022 at 12:14

1 Answer 1

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This is a long comment rather than an answer to the question above. The purpose of this post is to familiarize the readers with what lies one step ahead about the (Schauder) bases.

Let $X$ be a Banach space (not necessarily reflexive). Let $(e_n)$ be a Schauder basis of $X$ with a dual basic sequence $(f_n)$ as above. First of all, for every $f\in X^*$ $$f= \sum_n f(e_n) f_n $$ where the sum converges in the weak$^{*}$ topology of $X^{*}$. Let $L\subseteq X^*$ be the norm-closed linear span of $\{f_n\}$. $L=X^*$ if and only if $(e_n)$ is a shrinking basis for $X$, which means $$\lim_{n\to\infty}\sup_{x\in Z_n}\frac{|f(x)|}{\|x\|} = 0$$ where $Z_n =\overline{span}\{e_k:k\geq n\}$ is the norm-closure of the span of $\{e_k:k\geq n\}$. Please see the books below for details and further information:

  • Topics in Banach Space Theory by Albiac & Kalton
  • A Short Course on Banach Space Theory by Carothers
  • Sequences and Series in Banach Spaces by Diestel
  • An Introduction to Banach Space Theory by Megginson
  • Functional Analysis: An Introduction to Banach Space Theory by Morrison.
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  • $\begingroup$ There was a well-written answer (evidently deleted later) at the time I wrote this footnote. $\endgroup$
    – Onur Oktay
    Commented Feb 9, 2022 at 20:34
  • $\begingroup$ The post itself doesn't give a full reply but the books does, thank you $\endgroup$
    – p00pdigger
    Commented Mar 16, 2022 at 20:06

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