# What (classes of) Banach spaces are known to have Schauder basis?

Motivation: I am trying to see for what class of Banach spaces the following result is true:

There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with some property) and corresponding projections $P_n: X \to V_n$ such that
a) $\cup V_n$ is dense in $X$
b) $\sup_n ||P_n|| < \infty$

I know if the space has Schauder basis, then the result is automatic. Hence I would appreciate if you can let me know:
i) a reference for the above result (if it exists)
ii) positive results for large class of Banach spaces which has a Schauder basis

I have tried Googling and Wikipedia, but couldn't find systematic information about existence of Schauder basis. The only counterexample I found was given by Per H. Enflo.

Thank you! I apologize if there is any inappropriate etiquette in my post as I am relatively new to the forum.

• The property you define is usually called the $\Pi$ property. – Bill Johnson Jun 25 '13 at 20:12
• Most of the classical separable Banach spaces are known to have a Schauder basis. The book of Albiac and Kalton is good place to start. Singer's two volumes on bases in Banach spaces probably more than you would ever want to know. – Bill Johnson Jun 25 '13 at 20:15
• For a good expository article, read Casazza's contribution in the Handbook of the Geometry of Banach Spaces, vol. 1. – Bill Johnson Jun 25 '13 at 20:17
• Thanks Bill for pointing me to the references! I will check them out! I have also found this article of you which explains the $\pi$ property: link.springer.com/article/10.1007/BF02771464 – Clark Chong Jun 26 '13 at 3:19
• In [JRZ] you will find some results on when the $\pi$ property implies the existence of a finite dimensional decomposition and some results on the existence of bases, such as a separable complemented subspace of an $L+p$ space must have a Schauder basis. – Bill Johnson Jun 26 '13 at 15:55

The property you define is usually called the $\pi$ property.
In [JRZ], which you mentioned in a comment, you will find some results on when the $\pi$ property implies the existence of a finite dimensional decomposition and some results on the existence of bases, such as a separable complemented subspace of an $L_p$ space must have a Schauder basis.
• Thanks for the references given. I have read them except for Singer's two volumes and learnt a lot. I have the impression: the articles available are all trying to show the relation between the different properties -- in particular, whether they are distinct from each other and when a weaker property implies existence of basis. I couldn't find one which illustrate how to show a space has property $\pi$. So, here are my two questions: – Clark Chong Jul 5 '13 at 21:50
• 1) Do you know if anyone has considered the relation between Uniform Convex space and $\pi$ spaces? (J Diestel's "Geometry of Banach Spaces - Selected Topics" contains some info about the case when a UC space HAS a basis, while I am trying to see if there are classes of UC spaces known to have a basis) 2) In particular, do you know if $C_p$, the Banach space of Schatten-p class operators on a Hilbert space has property $\pi$? Thank you so much! – Clark Chong Jul 5 '13 at 21:54
• Casazza seems to have mentioned a "counterexample" in his handbook article - there exists a reflexive subspace of $l_p$ that fails Compact Approximation Property --> this would be a UC space which fails property $\pi$ – Clark Chong Jul 5 '13 at 21:56