So, I’ve shown that a finite-dimensional vector space $V$ is naturally isomorphic to its double dual $V’’$, ie $V\cong V’’$.
My question is why is this useful? When does this come in handy? What are the implications and consequences of this? Because I have never used this result, only proved it.
Thank you
Reflexivity of a vector space (ie. the condition that $V \cong V''$) is mostly useful (and nontrivial) in the infinite-dimensional setting when working with weak topologies. Most notably, the important Banach-Alaoglu theorem in functional analysis asserts that the closed unit ball of a reflexive space is weakly compact. This means that any bounded sequence in $V$ has a weakly converging subsequence. This can be used to prove the existence of minimizers of certain functionals in weakly closed subspaces of $V$ (the so-called calculus of variations method).
Indeed, suppose you want to know whether $F : M \subseteq V \to \mathbb{R}$ admits a minimizer, ie. a point $v \in M$ such that $F(v)\leq F(v')$ for all $v' \in M$. One method of proving such a point exists if $V$ is reflexive is by proving that (i) M weakly closed and (ii) F is coercive and weakly-lower semicontinuous. If both conditions are satisfied, then you can consider a minimising sequence $v_m$ of $F$ in $M$. By coercivity, you know $v_m$ is bounded ; by reflexivity, up to a subsequence, $v_m$ converges weakly to $v \in V$ ; by weak-closedness, $v \in M$, and finally by weak-lower semicontinuity, $F(v) \leq \lim\inf F(v_m) = \min_{M} F. $
The question of existence of minimizers to certain functionals is an important problem in analysis. As often minimisers of $F$ satisfy a partial differential equation linked to $F$, calculus of variations is one way of proving the existence (and sometimes regularity) of solutions to PDEs. For example, if $F(u) = \int_{\Omega} \sqrt{1+|\nabla u|^2}$ for $u$ in some function space $\Omega \subseteq \mathbb{R}^n \to \mathbb{R}$, then the minimiser (if it exists) satisfies the Laplace equation $\Delta u = 0$.