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Section 5.1 Bounded linear functionals

Remark 5.1.1.
In this chapter, we state claims in the setting of Frechet and Banach spaces and often we only provide proofs for the case of Banach spaces in order to avoid technicalities. Notice though that all statements, including the Hahn-Banach theorem, are actually valid in the much wider setting of Locally Convex (LC) spaces, namely spaces defined by families (not necessarily countable) of seminorms. The spaces \(L^p_c(\bR^n)\) (integrable functions with compact support), \(C^k_c(\bR^n)\) (\(C^k\) functions with compact support) and \(D^k(\bR^n)\) (see Section 5.5) are examples of spaces that are neither Banach nor Frechet but are LC. The same holds for weak and \(*\)-weak topologies.
Definition 5.1.2.
Let \(X\) be a Frechet vector space. We set \(X^*=\cB(X,\bR)\) and we call this space the dual of \(X\text{.}\) In finite dimension, the elements of \(X^*\) are called covectors. In infinite dimension, they are also often called linear functionals.
Let \(X\) be a finite-dimensional space. Then also \(X^*\) is finite-dimensional and \(\dim X=\dim X^*\text{.}\) Hence, in finite dimension, \(X\) and \(X^*\) are always isomorphic. Nevertheless, there is no canonical isomorphism and so these spaces cannot really be identified.

In order to understand this fact, take a basis \(e_1,\dots,e_n\) of \(X\text{.}\) The \(n\) "dual" covectors \(\eps^i\) defined by
\begin{equation*} \eps^i(e_j)=\delta^i_j \end{equation*}
are clearly linearly independent and generate \(X^*\text{.}\) The map \(e_i\to\eps^i\text{,}\) though, is not canonical in the following sense. Let \(f=Ae\) be a second basis of \(X\text{,}\) namely \(A\) is a \(n\times n\) matrix and
\begin{equation*} \begin{pmatrix}f_1\\ \vdots\\ f_n\\ \end{pmatrix} = A \begin{pmatrix}e_1\\ \vdots\\ e_n\\ \end{pmatrix} \end{equation*}
and denote by \(\eta^i\) the dual basis of \(f_1,\dots,f_n\text{.}\) Then, since \(\eta(f)=\mathbb 1_n\text{,}\) \(\eps(e)=\mathbb 1_n\) and \(f=Ae\text{,}\) we see that \(\eta A=\eps\text{,}\) namely
\begin{equation*} \eta = \eps A^{-1}. \end{equation*}
On the other side, the map \(e_i\to\eps^i\) writes matricially as \(e\to e^T\text{,}\) and under \(f=Ae\) we have that
\begin{equation*} f^T=e^T A^T. \end{equation*}
So, when we change basis via \(A\text{,}\) the dual bases change with \(A^{-1}\) while the bases obtained via tha naif map \(e_i\to\eps^i\) changes with \(A^T\text{,}\) so this identification is not preseved in general.

Notice, though, that this identification is preserved when one restricts to the class of matrices such that \(A^T=A^{-1}\text{.}\) These are called rotations and are the matrices that preserve the Euclidean scalar product. The fact that in this case the identification is canonical is reflected by the fact that it can it corresponds to the map \(e_i\to\langle e_i,\cdot\rangle\text{,}\) which is indeed preseved by isometries. This fact is at the core of the Riesz representation theorem in Hilbert spaces.

Recall that the codimension of a linear subspace \(M\subset X\) is the dimension of the quotient vector space \(X/M\text{.}\)
\(\ker\eta\) is closed because \(\eta\) is continuous. Now, suppose there were two linearly independent vectors \(v,u\) such that \(\eta(v)\eta(u)\neq0\text{.}\) Then \(w=\eta(u)v-\eta(v)u\) is non-zero but \(\eta(w)=0\)
This is just a particular case of the fact that \(\cB(X,Y)\) is a banach space under the operator norm as long as \(Y\) is a banach space -- the completeness of \(X\) is not needed for this result.

The result above is less surprising after realizing the following one.
Let \(\eta\in X^*\text{.}\) Since \(\eta\) is continuous on \(X\) and \(X\) is dense in \(\overline{X}\text{,}\) \(\eta\) extends continuously to \(X^*\text{.}\) Indeed, let \(x_n\in X\) such that \(x_n\to x\in \overline{X}\setminus X\text{.}\) Then \(x_n\) is Cauchy and so is \(\eta(x_n)\text{,}\) so that \(\eta(x_n)\to\ell\text{.}\) If \(x'_n\) is any other sequence with the same property, then \(x_n-x'_n\to0\) and so, by linearity, \(\eta(x_n)-\eta(x'_n)\to0\) and so \(\eta(x'_n)\to\ell\) as well. Hence we can define \(\bar \eta(x)=\ell\text{.}\) The map \(\bar \eta\) is linear since "the sum of limits is the limit of the sum" and it is continuous by construction. Moreover, since \(X\) is dense in \(\overline{X}\text{,}\) the \(\sup\) of \(|\eta(x)|/\|x\|\) on \(X\) and \(\overline{X}\) coincide and so \(\|\bar \eta\|_{\overline{X}^*}=\|\eta\|_{X^*}\text{.}\) Hence, \(\overline{X}^*\) and \(X^*\) are isometrically isomorphic.

This shows, for instance, that the dual space of \((C^\infty([0,1]),\|\cdot\|_{L^p([0,1])})\) coincides with the dual space of \(L^p([0,1])\) and so on. In other words, among normed spaces, one can limit the attention to duals of complete spaces.

The symmetry between \(\eta\) and \(v\) in the expression \(\eta_i v^i\) suggests that every vector \(v\) can be seen as a "co-covector", namely as an element of \(X^{**}\text{.}\) In other words, there is a linear map \(J:X\to X^{**}\) defined as \(J_v(\eta) = \eta(v)\text{.}\) We will prove in next section that this map is an isometric injection.

Before saying more about dual spaces, we need to know more about the elements that populate \(X^*\text{.}\) The fundamental tool in this matter is the Hahn-banach theorem, presented in next section.