Let \(n\) be a positive integer and \(F\) a field. Let \(W\) be the set of all vectors \((x_1, \ldots, x_n)\) in \(F^n\) such that \(x_1 + \cdots + x_n = 0\).
Prove that \(W^0\) consists of all linear functionals \(f\) of the form \[ f(x_1, \ldots, x_n) = c \sum_{j=1}^{n} x_j. \]
Show that the dual space \(W^*\) of \(W\) can be ‘naturally’ identified with the linear functionals \[ f(x_1, \ldots, x_n) = c_1 x_1 + \cdots + c_n x_n \] on \(F^n\) which satisfy \(c_1 + \cdots + c_n = 0\).
Solution
Write \(e_1,\ldots,e_n\) for the standard basis of \(F^n\) and define \[ \sigma(x_1,\ldots,x_n)=\sum_{j=1}^n x_j. \] Then \(W=\ker\sigma\).
1. The annihilator \(W^0\) is the one-dimensional space spanned by \(\sigma\).
Every linear functional on \(F^n\) has a unique expression \[ f(x_1,\ldots,x_n)=\sum_{j=1}^n c_jx_j. \] Suppose \(f\in W^0\). For each \(i<n\), the vector \(e_i-e_n\) belongs to \(W\). Therefore \[ 0=f(e_i-e_n)=c_i-c_n, \] so \(c_i=c_n\) for every \(i<n\). Thus all coefficients are equal to some scalar \(c\), and \[ f=c\sigma. \] Conversely, every \(c\sigma\) vanishes on \(W=\ker\sigma\). Hence \[ W^0=\{c\sigma:c\in F\}, \] which is exactly the description in part (a) [@HK71].
2. Every functional on \(W\) is the restriction of a functional on \(F^n\) whose coefficients sum to zero.
Let \(\lambda\in W^*\). Since \(W\) is a subspace of the finite-dimensional space \(F^n\), extend a basis of \(W\) to a basis of \(F^n\) and assign any value to the additional basis vector. This extends \(\lambda\) to some \[ f(x)=\sum_{j=1}^n c_jx_j \] on \(F^n\).
Put \[ s=\sum_{j=1}^n c_j, \qquad d_j=c_j-\frac{s}{n}. \] Hoffman–Kunze’s scalar fields have characteristic zero, so \(n\) is invertible in \(F\) [@HK71]. Then \[ \sum_{j=1}^n d_j=s-n\frac{s}{n}=0. \] The functional \[ g(x)=\sum_{j=1}^n d_jx_j \] differs from \(f\) by \[ g-f=-\frac{s}{n}\sigma, \] which lies in \(W^0\) by part (a). Therefore \(g|_W=f|_W=\lambda\). Thus every element of \(W^*\) has a representative with coefficient sum zero.
3. That representative is unique, giving the natural identification.
Let \[ L=\left\{f_c(x)=\sum_{j=1}^n c_jx_j:\sum_{j=1}^n c_j=0\right\} \subseteq(F^n)^*. \] Consider the restriction map \[ \rho:L\longrightarrow W^*,\qquad \rho(f)=f|_W. \] Step 2 proves surjectivity. If \(f_c\in\ker\rho\), then \(f_c\in W^0\). By part (a), all its coefficients equal a common scalar \(a\). Since their sum is zero, \[ na=0. \] Characteristic zero gives \(a=0\), hence \(f_c=0\). Therefore \(\rho\) is injective.
So restriction is a linear isomorphism \[ L\cong W^*, \] which is the claimed natural identification.