1.2 Subspaces

  • Definition. A vector (or linear) subspace of a vector space \(V\) over \(\F \) is a non-empty subset \(U\sub V\) which is closed under addition and scalar multiplication: whenever \(u,u_1,u_2\in U\) and \(\lambda \in \F \), then \(u_1+u_2\in U\) and \(\lambda u\in U\).

    In this case, we write \(U\leq V\).

    Say that \(U\) is trivial if \(U=\set {0}\) and proper if \(U\neq V\).

Of course, \(U\) is now a vector space in its own right using the addition and scalar multiplication of \(V\).

  • Exercise.2 \(U\sub V\) is a subspace if and only if \(U\) satisfies the following conditions:

    • (1) \(0\in U\);

    • (2) For all \(u_1,u_2\in U\) and \(\lambda \in \F \), \(u_1+\lambda u_2\in U\).

    This gives a efficient recipe for checking when a subset is a subspace.

2 Question 1 on sheet 1.

  • Examples. A good way to see that something is a vector space is to see that it is a subspace of some \(V^{\cI }\). That way, there is no need to verify all the tedious axioms (associativity, distributivity and so on).

    • (1) The set \(c:=\set {\text {real convergent sequences}}\leq \R ^{\N }\) and so is a vector space. This is part of the content of the Algebra of Limits Theorem in Analysis 1.

    • (2) Let \([a,b]\sub \R \) be an interval and set

      \begin{equation*} C^0[a,b]:=\set {f:[a,b]\to \R \st \text {$f$ is continuous}}, \end{equation*}

      the set of continuous functions.

      Then \(C^0[a,b]\leq \R ^{[a,b]}\). This is most of the Algebra of Continuous Functions Theorem from Analysis 1.