Basic sequence vs uniform boundedness of projections

Question

Let $\{e_n,e^*_n\}_{n=1}^\infty$ be a biorthogonal system in $\mathbb{X}\times\mathbb{X}^*$. The system $\{e_n\}_{n=1}^\infty$ is called a basic sequence if it is a Schauder basis of the subspace $E={\overline{{\rm span} \,\{e_n\}_{n=1}^\infty}}^{\mathbb X}$. A natural sufficient condition is to test whether the projection operators $$ S_N x=\sum_{n=1}^N e^*_n(x)e_n, \quad x\in\mathbb X,$$ are uniformly bounded in $\mathbb X$.

Is this condition also necessary? That is, is it true that $$ \{e_n\}_{n=1}^\infty \mbox{ is a basic sequence in $\mathbb X$ } \quad\Longleftrightarrow \quad\sup_N\|S_N\|_{\mathbb X\to\mathbb X}<\infty\;? \tag{*}$$ This is the case in simple examples, such as when $\{e_n\}$ is the canonical system in $\ell_\infty$ or the Haar system in $L_\infty$.

If the answer is NO, are there natural additional hypothesis on $\mathbb X$ (or on $\{e_n,e^*_n\}$) that ensure that ($*$) holds?

Answer 1

I think that I found an easy negative answer by adding a 1-dimensional subspace (sorry for not thinking enough before posting).

The details would be as follows. Assume that $\{e_n\}_{n=1}^\infty$ is a Schauder basis in $\mathbb X$, with dual functionals $e^*_n$, and say normalized $\|e_n\|_{\mathbb X}=\|e^*_n\|_{\mathbb X^*}=1$. Define a new space $\widehat{\mathbb {X}}=\mathbb {X}\oplus_1[x_0]$, with norm

$$\|(x,\lambda x_0)\|_{\widehat{\mathbb{X}}}=\|x\|_{\mathbb X}+|\lambda|.$$

Define a new biorthogonal system $\{\hat{e}_n,\hat{e}^*_n\}_{n=1}^\infty$ in $\widehat{\mathbb X}\times\widehat{\mathbb X}^*$ as follows:

$$ \hat{e}_n=(e_n,0) \quad \mbox{and}\quad \hat{e}^*_n(x,\lambda x_0)=e^*_n(x)+n\lambda.$$

Then $\{\hat{e}_n\}_{n=1}^\infty$ is a basic sequence in $\widehat{\mathbb X}$ (in fact a Schauder basis of $\mathbb{X}\oplus\{0\}$). However, the associated projection operators $\widehat{S}_N$ are not uniformly bounded, since

$$\widehat{S}_N(0,x_0)=\sum_{n=1}^N\hat{e}^*_n(0,x_0)e_n=\sum_{n=1}^Nne_n,$$ and therefore $$\|\hat{S}_N-\hat{S}_{N-1}\|\geq N.$$

In this example the dual functionals $\hat{e}^*_n$ have unbounded norms in ${\widehat{\mathbb X}^*}$ , but one could construct an example with bounded dual functionals, by setting $$ \hat{e}^*_n(x,\lambda x_0)=e^*_n(x)+\lambda,$$ provided that $$\|\widehat{S}_N(0,x_0)\|_{\widehat{\mathbb X}}=\|\sum_{n=1}^Ne_n\|_{\mathbb X}\to\infty,\quad \mbox{as }N\to\infty,$$ which happens, e.g. when $\mathbb{X}$ is super-reflexive.

One could still ask whether, for normalized biorthogonal systems $\{e_n,e^*_n\}_{n=1}^\infty\subset\mathbb{X}\times\mathbb{X}^*$ in spaces $\mathbb X$ such as $\ell_\infty$ or $L_\infty$, the equivalence in ($*$) may be true.