For Banach spaces, a one-dimensional subspace always has a closed complementary subspace.
2.
A given direct sum decomposition of X into complementary subspaces still specifies a projection, and vice versa.
3.
The Hodge index theorem says that the subspace spanned by " H " in " D " has a complementary subspace on which the intersection pairing is negative definite.
4.
It follows from the above conditions that, in the tangent space of an arbitrary point in " E ", the lifts " X " * span a two-dimensional subspace of "'horizontal "'vectors, forming a complementary subspace to the vertical vectors.
5.
I was wondering if anyone could perhaps point me to a ( simple-ish if possible please ! ) proof, or explain one to me, that 2 same-dimensional subspaces of a finite-dimensional vector space have a common complementary subspace-I've been staring at my page for about an hour or so now and as the time of day would suggest I'm getting nowhere-I previously managed to show that for any subspace U \ subset \ mathbb { F } ^ n, there is a subset I \ subset { 1, \, 2 \, . . . n } for which the subspace W = is complementary to U in \ mathbb { F } ^ n ( this is part ii, that was part i ! ), but I'm not sure to what extent that's related.
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