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

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

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.