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Linear subspace

A subset containing zero and closed under addition and scalar multiplication.

Let XX be a vector space over a field KK, and let Y⊆XY\subseteq X. The set YY is a linear subspace of XX if

  1. 0∈Y0\in Y,
  2. a+b∈Ya+b\in Y for all a,b∈Ya,b\in Y, and
  3. λa∈Y\lambda a\in Y for all λ∈K\lambda\in K and a∈Ya\in Y.

With the inherited operations, YY is itself a .

Examples
  • {0}\{0\} and XX are subspaces of XX.
  • In the vector space of all scalar sequences, ℓ1={x=(xn):∑n=1∞∣xn∣<∞}\ell^1=\{x=(x_n):\sum_{n=1}^\infty|x_n|<\infty\} is a subspace.
  • The continuous functions C([a,b])C([a,b]) form a subspace of all functions [a,b]→K[a,b]\to K.