In [1], Böttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space \documentclass[12pt]{minimal}
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\begin{document}$$H, \{e_{j}\}_{j=1}^{\infty}$$\end{document} is an orthonormal basis of H and Pn is the orthogonal projection onto the span of \documentclass[12pt]{minimal}
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\begin{document}$$\{e_{j}\}_{j=1}^{n}$$\end{document}, then for each \documentclass[12pt]{minimal}
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\begin{document}$$k \in {\mathbb{N}}$$\end{document}, the sequence \documentclass[12pt]{minimal}
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\begin{document}$$\{s_{k}(P_{n}TP_{n})\}$$\end{document} converges to sk(T), where for a bounded operator A on H, sk(A) denotes the kth approximation number of A, that is, sk(A) is the distance from A to the set of all bounded linear operators of rank at most k − 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {Pn} and {Qn} are sequences of bounded linear operators on X and Y, respectively, such that \documentclass[12pt]{minimal}
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\begin{document}$$\|P_n\| \|Q_n\| \leq 1 $$\end{document} for all \documentclass[12pt]{minimal}
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\begin{document}$$n \in {\mathbb{N}}$$\end{document} and {QnTPn} converges to T under the weak operator topology, then \documentclass[12pt]{minimal}
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\begin{document}$$\{s_{k}(Q_{n}TP_{n})\}$$\end{document} converges to sk(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of \documentclass[12pt]{minimal}
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\begin{document}$$s_{k}(Q_{n}TP_{n})$$\end{document} to sk(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {Pn} and {Qn} on X and Y respectively such that (i) \documentclass[12pt]{minimal}
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\begin{document}$$Q_{n}TP_{n}$$\end{document} is compact, (ii) \documentclass[12pt]{minimal}
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\begin{document}$$\|P_{n}\| \|Q_{n}\| \leq 1 $$\end{document} and (iii) \documentclass[12pt]{minimal}
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\begin{document}$$\{Q_{n}TP_{n}\}$$\end{document} converges to T in the weak operator topology, then \documentclass[12pt]{minimal}
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\begin{document}$$\{s_k(Q_{n}TP_{n})\}$$\end{document} converges to sk(T) if and only if \documentclass[12pt]{minimal}
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\begin{document}$$s_{k}(T) = s_{k}(T^\prime)$$\end{document}. This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.
