Editing Linear subspace (section)

== Properties of subspaces ==
From the definition of vector spaces, it follows that subspaces are nonempty, and are [[Closure (mathematics)|closed]] under sums and under scalar multiples.<ref>{{Harvtxt|MathWorld|2021}} Subspace.</ref> Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set ''W'' is a subspace [[if and only if]] every linear combination of [[finite set|finite]]ly many elements of ''W'' also belongs to ''W''.
The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.

In a [[topological vector space]] ''X'', a subspace ''W'' need not be topologically [[closed set|closed]], but a [[finite-dimensional]] subspace is always closed.<ref>{{harvtxt|DuChateau|2002}} Basic facts about Hilbert Space — class notes from Colorado State University on Partial Differential Equations (M645).</ref> The same is true for subspaces of finite [[codimension]] (i.e., subspaces determined by a finite number of continuous [[linear functional]]s).