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Linear Algebra Companion: A supplement to undergraduate linear algebra enhanced by computation

Thomas R. Shemanske

Section 2.1 Vector spaces

It may be useful to recall the definition of a vector space. Clicking the link will drop down the definition.
Some very common examples of vector spaces over a field \(F\) (e.g., \(F=\Q, \R\text{,}\)or \(\C\)).
We continue with somewhat more sophisticated examples of vector spaces. These are vector spaces of functions which are especially important when linear algebra is applied to analysis. You have aleady studied some of these spaces when you took calculus.
  • Let
    \begin{equation*} V=\cF(\R)=\{ f:\R\to \R\}, \end{equation*}
    be the set of all functions from \(\R\) to \(\R\text{.}\) This is the underlying set which we want to make \(V\) into a vector space. To do so, we need to define vector addition and scalar multiplication. For \(f,g \in \cF(\R)\) and \(\lambda \in \R\text{,}\) define \(f+g\) and \(\lambda f\) by:
    \begin{gather*} (f+g)(x) := f(x) + g(x) \text{ for all } x\in \R\\ (\lambda f)(x) := \lambda f(x) \text{ for all } x\in \R. \end{gather*}
    We leave it as an exercise to show that \(\cF(\R)\) is a vector space over \(\R.\)
  • Let
    \begin{equation*} C(\R)=\{f\in \cF(\R) \mid f \text{ is continuous }\}\text{.} \end{equation*}
    Is \(C(\R)\) a subspace of \(\cF(\R)\text{?}\) What do we need to check?
    Certainly, the zero function \(0(x) = 0\) for all \(x \in \R\) is a continuous function. Is the sum of two continuous functions continuous and is a constant times a continuous function continuous? Wait!? Aren’t those theorems one proves in calculus?
    The take away here is that sometimes even verifying the closure axioms can be nontrivial.
  • Generalizing the above, we define
    \begin{equation*} C^n(\R)= \{f \in \cF(\R)\mid f^{(n)} \text{ is continuous.}\} \end{equation*}
    Implicit in this definition (again a theorem from calculus) is that if \(f\in C^{n}(\R)\text{,}\) then \(f, f', f'', \cdots, f^{(n)}\) are all continuous. We see that we have a nested sequence of subspaces:
    \begin{equation*} C^{(n)}(\R) \subset C^{(n-1)}(\R) \subset \cdots \subset C(\R)\subset \cF(\R). \end{equation*}
  • Finally we define \(C^\infty(\R)\) to be the elements of \(\cF(\R)\) which are in \(C^n(\R)\) for all \(n\ge 1\text{.}\) Said symbolically,
    \begin{equation*} C^\infty(\R) = \bigcap_{n=1}^\infty C^{(n)}(\R). \end{equation*}
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