In linear algbera we learn that a vector space $V$ is closed under addition and scalar multiplication such that any vector $v$ in $V$ is a linear combination of some basis set $S$. In discrete math we learn that a function is recursively defined when there exists some basis $f(0)=a$ for $f(n) = kf(n-1)$.
These concepts seem similar but is it right to say that a vector space is recursively defined by its basis? And if not can a vector space even be recursively defined? My gut feeling is that closure would not hold, but I am still new to mathematics, so any help would be appreciated!
Edit: For clarification to whpowell96's point, for this problem I only wish to consider every vector space with a basis, such that its basis is also a recursive basis.
