TITLE: Example where $(V^*)^*\neq V$?
QUESTION [5 upvotes]: Let $V$ be a vector space. This answer gives a nice explanation of why the "dual dual space" of $V$, i.e., $(V^*)^*$, is isomorphic to $V$ if $\dim V<\infty.$ 

Can someone give an example where $\dim V=\infty$ and $(V^*)^*\ncong V$? I'm having a hard time imagining this. 
Additionally, must $V$ always be at least isomorphic to a subspace of $(V^*)^*$? It seems that the answer must be yes, since we can always define $\xi_{v∈ V}:V^*\to\mathbb R, \omega\mapsto\xi_v(\omega):=\omega(v).$

REPLY [3 votes]: For the first question. For every vector space $V$ it is true that 
$$
\dim(V)\leq \dim(V^*)
$$
with equality if and only if $V$ is finite dimensional (see Theorem 3.12 in Advanced Linear Algebra by Steven Roman). Thus, if $V$ is infinite dimensional, then $V\not\cong V^{**}$, so every infinite dimensional vector space serves you as an example.
For question 2: Yes.