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Yes, it is easy to prove that "a basis exists" (almost by definition depending on how exactly you define dimension) and of course it's legal to use it in proofs
by zodiac 9y ago
Yes, it is easy to prove that "a basis exists" (almost by definition depending on how exactly you define dimension) and of course it's legal to use it in proofs, but I think proofs that avoid using this fact directly are more elegant
- Bromskloss 9y ago> almost by definition depending on how exactly you define dimension What definition of _dimension_ is there that does not rely upon the existence of a base?
- zodiac 9y agoDoes "maximum size of a linearly independent set" count? You would have to prove something about spanning sets to get to the existence of a basis
- contravariant 9y agoYou could define the dimension to be the supremum of the cardinality of all linearly independent sets. In the infinite case this does not trivially give you a basis as 1) the supremum could be strictly larger than the cardinality of all linearly independent sets and 2) adding an extra vector to an infinite linearly independent set doesn't increase it's cardinality, hence there is no reason for the basis to span the entire space. You could also take the infimum of all sets that span the entire space, but you run into similar problems.