Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).

Question

[LINEAR FUNCTIONALS] In \(R^{n}\) with the usual inner product, a linear functional \(\ell:\) \(\mathbb{R}^{n} \rightarrow \mathbb{R}\) is just a linear map into the reals (in a complex vector space, it maps into the complex numbers \(\mathbb{C}\) ). Define the norm, \(\|\ell\|\), as
\[
\|\ell\|:=\max _{\|x\|=1}|\ell(x)| .
\]
a) Show that the set of linear functionals with this norm is a normed linear space.
b) If \(v \in \mathbb{R}^{n}\) is a given vector, define \(\ell(x)=\langle x, v\rangle\). Show that \(\ell\) is a linear functional and that \(\|\ell\|=\|v\|\).
c) [REPRESENTATION OF A LINEAR FUNCTIONAL] Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).
d) [EXTENSION OF A LINEAR FUNCTIONAL] Let \(U \subset \mathbb{R}^{n}\) be a subspace of \(\mathbb{R}^{n}\) and \(\ell\) a linear functional defined on \(U\) with norm \(\|\ell\|_{U}\). Show there is a unique extension of \(\ell\) to \(\mathbb{R}^{n}\) with the property that \(\|\ell\|_{\mathbb{R}^{n}}=\|\ell\|_{U}\).
[In other words define \(\ell\) on all of \(\mathbb{R}^{n}\) so that on \(U\) this extended definition agrees with the original definition and so that its norm is unchanged].