Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis which is not another. For example, given that
\begin{equation}\text{symmetric basis }\Rightarrow\text{ subsymmetric basis }\Rightarrow\text{ unconditional basis }\Rightarrow\text{ basis}\end{equation}
we want to prove that the reverse implications do not hold, i.e. that
\begin{equation}\tag{$\*$}\text{basis }\nRightarrow\text{ unconditional basis }\nRightarrow\text{ subsymmetric basis }\nRightarrow\text{ symmetric basis}.\end{equation}
The above nonimplications are well-known.


However, for every type of basis, there is a corresponding property of Banach spaces for *admitting* that type of basis. To continue the above example, we have:
\begin{equation}\begin{array}{c}\text{admits a}\\\text{symmetric basis}\end{array}\Rightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\Rightarrow\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\Rightarrow\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\end{equation}
Let us now consider the reverse nonimplications.
\begin{equation}\tag{$\*\*$}\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\nRightarrow\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{symmetric basis}\end{array}\end{equation}
Proving ($\*\*$) is more difficult than proving ($\*$). However, the first two nonimplications in ($\*\*$) are already known. For instance, $L\_1[0,1]$ admits a basis but not an unconditional one. $L\_p[0,1]$, $p\in(1,2)\cup(2,\infty)$, admits an unconditional basis but not a subsymmetric one.


Garling (1968) exhibited an example of a basis which is subsymmetric but not symmetric. His basis was a modification of the canonical basis of the Lorentz sequence space $d(w,p)$ with $w=(n^{-1/2})\_{n=1}^\infty$ and $p=1$. In a Lorentz sequence space, the norm of a vector is taken by considering the supremum over all permutations of an $\ell\_p$ basis, with weights applied in a certain way. If instead you consider the supremum over all *subsequences* of an $\ell\_p$ basis (with weights similarly applied), you get a Garling-like basis.


It is known that $d(w,p)$ admits a unique symmetric basis. Using similar arguments, we can show that Garling's space (and indeed a wider class of spaces based on Garling's construction) admits a unique subsymmetric basis. Consequently, we have examples of spaces which admit a subsymmetric basis but not a symmetric one. This proves the third nonimplication in ($\*\*$), and brings me to my first question.


**Question 1.** Was the third nonimplication in ($\*\*$) already known? In other words, was there already known an example of a Banach space which admits a subsymmetric basis but not a symmetric one?


I am hoping the answer is no, as it feels good to contribute something new to the math community : )


The next natural question is to expand ($\*\*$) to include other types of bases. For example, we have the following:
\begin{equation}\text{unconditional basis }\nRightarrow\text{ greedy basis }\nRightarrow\text{ subsymmetric basis}\end{equation}
Accordingly, we can show
\begin{equation}\tag{$\*\*\*$}\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{greedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\end{equation}
Dilworth/Kalton/Kutzarova (2003) proved that $c\_0$ is the unique $\mathcal{L}\_\infty$-space admitting a quasigreedy basis, and hence the unique $\mathcal{L}\_\infty$-space admitting a greedy basis. So, if we consider any $\mathcal{L}\_\infty$ space with an unconditional basis but which is not isomorphic to $c\_0$ then we prove the first nonimplication in ($\*\*\*$). The Haar basis for $L\_p[0,1]$, $p\in(1,2)\cup(2,\infty)$, is greedy, which proves the second nonimplication in ($\*\*\*$).


We can further subdivide greedy bases as follows.
\begin{equation}\text{greedy basis }\Rightarrow\text{ almost greedy basis }\Rightarrow\text{ quasigreedy basis }\Rightarrow\text{ basis}\end{equation}
and
\begin{equation}\text{basis }\nRightarrow\text{ quasigreedy basis }\nRightarrow\text{ almost greedy basis }\nRightarrow\text{ greedy basis}\end{equation}
leading to the conjecture
\begin{equation}\tag{$\*\*\*\*$}\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{quasigreedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits an}\\\text{almost greedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{greedy basis}\end{array}\end{equation}
The $(\oplus\_{n=1}^\infty\ell\_\infty^n)\_{\ell\_1}$ counterexample proves the first nonimplication in ($\*\*\*\*$), and Gogyan's construction (2010) of an almost greedy basis for $L\_1[0,1]$ proves the third. However, we have the following question


**Question 2.** Does the second nonimplication in ($\*\*\*\*$) hold? In other words, can we find an example of a Banach space admitting a quasigreedy basis but not an almost greedy basis?


I suspect that the answer to Questions #2 may be "no," as the connections between almost/quasi/fully greedy bases seem to be closer than between other types of bases. For example, Albiac/Ansorena (2015) showed that if a basis is both 1-quasigreedy and 1-almost-greedy then it is 1-greedy. Meanwhile, Dilworth/Kalton/Kutzarova (2003) proved that $c\_0$ is the only infinite-dimensional Banach space admitting a unique quasigreedy basis, which ruins any chance of using on ($\*\*\*\*$) the method I used in proving the third nonimplication in ($\*\*$).


Finally:


**Question 3.** What are some other nonimplications of the form of ($\*\*$)/($\*\*\*$)/($\*\*\*\*$) which are known, or unknown?


**Question 4.** Are Questions #1-3 interesting?


Thanks!



Non-equivalence of admitting different types of bases in Banach spaces
For Q2, consider the original Tsirelson space. It has an unconditional, then quasi greedy, basis. However, it does not contain a democratic basis (see Remark 5.8 in DKK2003).


