A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces. Let K be a field, V be a space over K. Then W is a if, The zero vector,0, is in W. If u and v are elements of W, then the sum u + v is an element of W, take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Proof, Given u and v in W, Thus, u + v is an element of W, too. Given u in W and a c in R, if u = again. Thus, cu is an element of W too, example II, Let the field be R again, but now let the vector space be the Cartesian plane R2. Take W to be the set of points of R2 such that x = y, then W is a subspace of R2. Proof, Let p = and q = be elements of W, then p + q =, since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. Let p = be an element of W, that is, a point in the plane such that p1 = p2, then cp =, since p1 = p2, then cp1 = cp2, so cp is an element of W. In general, any subset of the coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace. Geometrically, these subspaces are points, lines, planes, and so on, example III, Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C be the subset consisting of continuous functions, then C is a subspace of RR. Proof, We know from calculus that 0 ∈ C ⊂ RR and we know from calculus that the sum of continuous functions is continuous. Again, we know from calculus that the product of a continuous function, example IV, Keep the same field and vector space as before, but now consider the set Diff of all differentiable functions. The same sort of argument as before shows that this is a subspace too, examples that extend these themes are common in functional analysis. A way to characterize subspaces is that they are closed under linear combinations, in a topological vector space X, a subspace W need not be closed in general, but a finite-dimensional subspace is always closed. The same is true for subspaces of finite codimension, i. e. determined by a number of continuous linear functionals
The vectors u and v are a basis for this two-dimensional subspace of R3.
In R3, the intersection of two distinct two-dimensional subspaces is one-dimensional