Does there exist an infinite dimensional real normed linear space which can be written as a countable union of one dimensional subspaces?

Question

Does there exist an infinite dimensional real NLS which can be written as a countable union of one dimensional subspaces ?

I can only tell that if there is a NLS such that it is possible then it cannot be a Banach space by Baire category theorem ( as any one dimensional subspace is closed and being a proper subspace , has empty interior ) .

Please help . Thanks in advance

Answer 1

Every one dimensional subspace of a real normed vector space intersects the unit sphere in two points. Thus, a countable union intersects the unit sphere in at most countably many points and cannot give the full space (Exercise: show that that the unit sphere is not countable)