A question about two complemented subspaces I would like to know about the differences between the following two complemented spaces.
$(1):$ If $V$ is a linear space(dim $V =\infty$ or finite), $W$ (dim $W=\infty$ or finite) is a subspace of $V$,
then there exists a subspace $W_1$ such that $V=W\oplus W_1$.
$(2):$ If $E$ is a Banach space, $G$ is a fnite-dimensional subspace of $E$,
then there exists a subspace $L$ such that $E=G\oplus L$.
What are the differences between $W_1$ and $L$?
If there is no difference between $W_1$ and $L$ , why does the second case require $G$ to be finite dimension?
Thanks!
 A: I think the real point is that when you write  $V = W \oplus W_1$ in the context of a linear space, you just require $W$ and $W_1$ to be linear subspaces whose span is $V$ and whose intersection is $\{0\}$.  When you write this in the context of a Banach space, you require $W$ and $W_1$ to be closed linear subspaces.  In the context of linear spaces, all subspaces are complemented, i.e. for any linear subspace $W$ of $V$ there is always a $W_1$.  In the context of Banach spaces, not all closed linear subspaces are complemented.
A: In both cases, $W_1$ and  are complements of  and , respectively, meaning that they are subspaces of  and , respectively, such that  and  can be expressed as the direct sum of  and $_1$, and  and , respectively.
The main difference between $_1$ and  is that $_1$ is not necessarily a Banach space, while  is required to be a Banach space. In other words,  is a complemented subspace of a Banach space , and it must inherit the Banach space structure from . On the other hand, 1 is simply a subspace of , which may or may not be a Banach space.
Another difference is that in the first case,  is a subspace of , which may be infinite-dimensional, while in the second case,  is a finite-dimensional subspace of .
