91Ó°ÊÓ

If \(V\) is spanned by a finite set, it is said to be finite-dimensional. The dimension of V, written as \(\dim V\), is the number of vectors in abasis for \(V\). The dimension of the zero-vector space\(\left\{ 0 \right\}\)is defined to be zero. If \(V\) is not spanned by a finite set, it is said to beinfinite-dimensional.

Consider \(\dim {\mathop{\rm P}\nolimits} = k < \infty \). Since \({{\mathop{\rm P}\nolimits} _n}\) is a subspace of \[{\mathop{\rm P}\nolimits} \] for all \(n\), and \(\dim k - 1 = k\), then \(\dim {{\mathop{\rm P}\nolimits} _{k - 1}} = \dim {\mathop{\rm P}\nolimits} \). This implies that \({{\mathop{\rm P}\nolimits} _{k - 1}} = {\mathop{\rm P}\nolimits} \), which is untrue. For example, \({\mathop{\rm p}\nolimits} \left( t \right) = {t^k}\) is in \[{\mathop{\rm P}\nolimits} \] but not in \[{{\mathop{\rm P}\nolimits} _{k - 1}}\]. Therefore, the dimension of \[{\mathop{\rm P}\nolimits} \] cannot be finite.

Thus, space \[{\mathop{\rm P}\nolimits} \] of all polynomials is infinite-dimensional.