# Adjoint system associated to Schauder basis

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.

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

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: