- #1
mathboy
- 182
- 0
Maximal subspace
Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.
I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?
Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.
I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?
Last edited: