Why are all finite-dimensional subspaces of a Banach space closed?
I hear you shouting this question to the uncaring heavens.
At last, your curiosity will be satisfied.
I will actually prove something stronger: let M be a closed subspace of a Banach space X (over a field K). Let x be any vector in X not lying in M1. Let N be the direct sum of M and the subspace Kx spanned by x. I put it to you that N is also closed. If N is all of X, then there's nothing to prove. So let Y=X\N, and suppose Y is nonempty. Since M is closed, by a corollary to the Hahn-Banach Theorem, for every y in Y we can find a functional fy in X* with fy(y)=1 and fy(z)=0 for all z in M. Likewise we can find a functional g with g(x)=1 and g(z)=0 for all z in M. Now let hy=fy-fy(x)g. Then hy(z)=0 for all z in M, and hy(x)=fy(x)-fy(x)g(x)=0, so hy(z)=0 for all z in N: N is contained in the kernel ker hy, for every y. Thus N is contained in the intersection of all the ker hy, as y ranges over Y: this is an intersection of closed sets, hence a closed set. And I put it to you that this intersection is exactly N, which will complete the proof. For suppose there existed y' in Y which lay in this intersection. Then we'd have hy(y')=0 for all y. But then fy(y')-fy(x)g(y')=0, or fy(y'-g(y')x)=0, so y'-g(y')x lies in ker fy for all y in Y; therefore it's in the intersection of all such kernels. But the intersection of all the ker fy is truly contained in N: for any vector in Y=X\N, we constructed an fy which was 1 at that vector. Hence y'-g(y')x lies in N, and since x lies in N, y' lies in N, a contradiction. QED
Now, the zero subspace {0} is closed. By the above, then, any one-dimensional subspace is closed: it's the direct sum of {0} with Kx for some x. By induction, therefore, all finite-dimensional subspaces are closed.
This seemed like the natural way, to me, to prove this when it was assigned as homework in the winter: if you need a vector subspace for some reason, look at kernels of linear transformations. But for some reason, analysts don't find it especially clear. Your mileage, as the old saw goes, may vary.
***
1. I left out the last clause originally. Silly me. I tacitly assumed all along that x did not lie in M but forgot to specify that. (22 June)