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.
Best Answer
The property you define is usually called the $\pi$ property.
For a good expository article, read Casazza's contribution in the Handbook of the Geometry of Banach Spaces, vol. 1.
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 "Bases in Banach Spaces" probably contains more than you would ever want to know.
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.