Assuming the axiom of choice, every vector space has a basis. In particular, every subspace have a basis. However assuming the axiom of choice does not hold, there are spaces without a basis. Of course that if V is a vector space without a basis it may have a subspace which has a basis, e.g. a span of a single vector.
Does a subspace always have a basis?
Proposition. Every nonzero subspace V of Rn has a basis. Proposition. Any two bases for V have the same number of vectors.
Does every subspace have an orthogonal basis?
An orthogonal set of unit vectors is called an orthonormal basis, and the Gram-Schmidt procedure and the earlier representation theorem yield the following result. Every subspace W of Rn has an orthonormal basis.
