Properties of Subspaces
A way to characterize subspaces is that they are closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of (finitely many) elements of W also belongs to W. Conditions 2 and 3 for a subspace are simply the most basic kinds of linear combinations.
A subspace W of X need not be closed in general, but a Euclidean subspace is always closed.
Read more about this topic: Linear Subspace
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—John Locke (1632–1704)