Weak Topology on an Infinite-Dimensional Hilbert Space

In summary, the weak topology on an infinite-dimensional Hilbert space is induced by the weak convergence of sequences, while the norm topology is induced by the norm function. The weak topology allows for a more natural way to define convergence and continuity for linear functionals and operators on infinite-dimensional Hilbert spaces. While weak convergence does not necessarily imply strong convergence in an infinite-dimensional space, a weakly convergent sequence can also converge in the norm topology. The weak-* topology, a special case of the weak topology, is important in studying the properties of functionals on a Hilbert space.
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Prove that the weak topology on an infinite-dimensional Hilbert space is non-metrizable.
 
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By way of contradiction, suppose that there exists a metrizable, infinite-dimensional Hilbert space ##H##. Let ##d## be a metric on ##H## inducing the weak topology. For every ##n \ge 1##, the origin belongs to the weak closure of the sphere in ##H## of radius ##n##. So there is a sequence ##(x_n)\subset H## with ##\|x_n\| = n## such that ##d(x_n,0) < n^{-1}##. Thus ##x_n## weakly converges to ##0##, even though ##(x_n)## is unbounded. (##\rightarrow\leftarrow##)
 
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