which case the weak* topology is metrizable on norm-bounded subsets of . If a normed space has a dual space that is separable (with respect to the dual-norm topology) then is necessarily separable. If is a Banach space, the weak-* topology is not metrizable on all of unless is finite-dimensional. Examples Hilbert spaces Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space . Strong convergence of a sequence to an element means that as . Here the notion of convergence corresponds to the norm on . In contrast weak convergence only demands that for all functions (or, more typically, all f in a dense subset of such as a space of test functions, if the sequence {ψk} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in . For example, in the Hilbert space , the sequence of functions form an orthonormal basis. In particular, the (strong) limit of as does not exist. On the other hand, by the Riemann–Lebesgue lemma, the weak limit exists and is zero. Distributions One normally obtains spaces of distributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on ). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as . Thus one is led to consider the idea of a rigged Hilbert space. Weak topology induced by the algebraic dual Suppose that is a vector space and X# is the algebraic dual space of (i.e. the vector space of all linear functionals on ). If is endowed with the weak topology induced by X# then the continuous dual space of is , every bounded subset of is contained in a finite-dimensional vector subspace of , every vector subspace of is closed and has a topological complement. Operator topologies If and are topological vector spaces, the space of continuous linear operators may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space to define operator convergence . There are, in general, a vast array of possible operator topologies on , whose naming is not entirely intuitive. For example, the strong operator topology on is the topology of pointwise convergence. For