Exercise 2.6.14

We use the notation in the Hint. To prove that $\alpha =\{f_{k+1},f_{k+2},\dots ,f_n\}$ is a basis for $W^0$, we should only need to prove that span$(\alpha )=W^0$ since by $\alpha \subset {\beta }^{\text{∗}}$ we already know that $\alpha$ is an independent set. Since $W^0\subset V^{\text{∗}}$, every element $f\in W^0$ we could write $f={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{a}_i{f}_i$. Next since for $1\le i\le k$ $x_i$ is an element in $W$, we know that

So actually we have $f={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=k+1}^n{a}_i{f}_i$ is an element in span$(\alpha )$. And finally we get the conclusion by