Space and it's space of linear functions are of same dimensionality

Question

In the following book on the page 141, https://www.springer.com/gp/book/9781468470550 there is written that: "Space and it's space of linear functions have same dimensionality". Why is that true? Where can i find this result with a proof?

Answer 1

This is not true in general, but it is true for finite-dimensional vectors spaces. Let $V$ be such a space and let $k$ be the field of scalars. If $(v_1,\ldots,v_n)$ is a basis of $V$ and if $\alpha_i\colon V\longrightarrow k$ is such that$$(\forall j\in\{1,\ldots,n\}):\alpha_i(v_j)=\begin{cases}1&\text{ if }j=i\\0&\text{ otherwise,}\end{cases}$$then it is not hard to prove that $(\alpha_1,\ldots,\alpha_n)$ is a basis of $V^*$. Every Linear Algebra textbook proves this statement.

Answer 2

I can't believe you've quoted the book accurately. It should say "A finite dimensional vector space and its space of linear functionals have the same dimension." Several differences there, the most important being the addition of "finite dimensional" and "functionals" instead of "functions" (and the most irritating being "its" in place of "it's"...)

Say $V$ is a finite dimensional vector space over the field $F$. Say $b_1,\dots,, b_n$ is a basis for $V$. For each $k$ there exists a linear $L_k:V\to F$ such that $$L_k b_j=\begin{cases}1,&(j=k), \\0,&(j\ne k).\end{cases}$$You can easily verify that $L_1,\dots,L_n$ is a basis for the space of linear functionals on $V$.