Definition of finite dimensional decomposition of Banach space

Question

The question is in the title.

What is the definition of finite dimensional decomposition of Banach space?

I have been looking around for a while and can't find anything! Thanks

Answer 1

Let $X$ be a Banach space. A sequence $\{X_n\}_{n=1}^\infty$ of finite-dimensional subspaces of $X$ is called a finite-dimensional decomposition of $X$ if every $x\in X$ has a unique representation of the form $x=\sum_{n=1}^\infty x_n$ with $x_n\in X_n$ for every $n\in \mathbb{N}$.

This is Definition 4.31 on page 198 of Banach Space Theory: The Basis for Linear and Nonlinear Analysis by Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler.