Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$, under the operations of intersection and weak sum (see $\S 1$ for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$. Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8]. In the main theorem below, we extend a result of Lachlan from the lattice $\mathscr{E}$ of r.e. sets to $\mathscr{L}(V_\propto)$. We define hyperhypersimple vector spaces, discuss some of their properties and show if $A, B \in \mathscr{L}(V_\propto)$, and $A$ is a hyperhypersimple subspace of $B$ then there is a recursive space $C$ such that $A + C = B$. It will be proven that if $V \in \mathscr{L}(V_\propto)$ and the lattice of superspaces of $V$ is a complemented modular lattice then $V$ is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.