Schur's Property
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High Quality Content by WIKIPEDIA articles! In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm. When we are working in a normed space X and we have a sequence (xn) that converges weakly to x (see weak convergence), then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to x in norm? Suppose that we have a normed space (X, · ), x an arbitrary member of X, and (xn) an arbitrary sequence in th...
High Quality Content by WIKIPEDIA articles! In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm. When we are working in a normed space X and we have a sequence (xn) that converges weakly to x (see weak convergence), then a natural question arises. Does the sequence converge in perhaps a more desirable manner? That is, does the sequence converge to x in norm? Suppose that we have a normed space (X, · ), x an arbitrary member of X, and (xn) an arbitrary sequence in the space. We say that X has Schur's property if (xn) converging weakly to x implies that lim_{ntoinfty} Vert x_n - xVert = 0 . In other words, the weak and strong topologies share the same convergent sequences. Note however that weak and strong topologies are always distinct in infinite-dimensional space.
