Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear maps from $V$ to $\mathbb{R}$ (or $\mathbb{C}$) and $U$ ranges over all open subsets of $\mathbb{R}$ (or $\mathbb{C}$).
If $V$ is infinite dimensional, one may define two (different?) topologies on $V$, one is repeating the same definition as in the finite case, another is to give $V$ the weak topology determined by all finite-dimensional subspaces; that is, a subset $C\subset V$ is closed if and only if $C\cap W$ is closed for every finite dimensional subspace $W$ of $V$.
My question is, are these two definitions the same? If not, which one is finer, and which one is more natural and frequently used?