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In this paper the weak topology on a normed space is studied from theviewpoint of infinite-dimensional topology. Besides the weak topology on anormed space $X$ (coinciding with the topology of uniform convergence on finitesubsets of the dual space $X^*$), we consider the topology $c$ of uniformconvergence on compact subsets of $X^*$. It is known that this topologycoincides with the weak topology on bounded subsets of $X$, but unlike to thelatter has much better topological properties (e.g., is stratifiable).  We prove that for normed spaces $X,Y$ with separable duals the spaces$(X,weak)$, $(Y,weak)$ are sequentially homeomorphic if and only if $\mathcalW(X)=\mathcal W(Y)$, where $\mathcal W(X)$ is the class of topological spaceshomeomorphic to closed bounded subsets of $(X,weak)$. Moreover, if $X,Y$ areBanach spaces which are isomorphic to their hyperplanes and have separaleduals, then the spaces $(X,weak)$ and $(Y,weak)$ are sequentially homeomorphicif and only of the spaces $(X,c)$ and $(Y,c)$ are homeomorphic. To prove thisresult, we show that for a normed space $X$ which is isomorphic to itshyperpane and has separable dual, the space $(X,c)$ (resp. $(X,weak)$) is(sequentially) homeomorphic to the product $B\times\mathbb R^\infty$ of theweak unit ball $B$ of $X$ and the linear space $\mathbb R^\infty$ withcountable Hamel basis and the strongest linear topology.

On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence