there is one question bothering me for quite a while now. Let $a_{n},b_{n}\in L^{2}:a_{n}\stackrel{L^{2}}{\rightharpoonup} a\in L^{2} $ weakly $ b_{n}\stackrel{L^{2}}{\rightarrow} b \in L^{2}$ strongly and $a_{n}\cdot b_n\in L_{2}$ weakly and let all the sequences, also the product sequence, be bounded in $L^{2}$, that means w.l.o.g. $a_{n}\cdot b_n\stackrel{L^{2}}{\rightharpoonup}c\in L^{2}$ weakly I’d like to prove that $a_{n}\cdot […]

Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces. When is $\operatorname{Ran}(T)$ a closed subspace? What theorems are there? Thanks ðŸ™‚

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