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I need to prove that $0 \in \bar{S_{X}}^{w}$ where $S_{X}=\{x\in X : ||x||=1\}$ and $X$ is a normed linear vector space. My idea is take a weak neighbourhood of $0$ in X and show that this weak neighbourhood intersects with $S_{X}$ but from here i don't have idea how to continue. how can i develop this idea? (in case that this idea would be correct)

LMOTRU
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  • what is $X$? If $X$ is an infinite dimensional Hilbert space, then any sequence of orthonormal vectors converges weakly to $0$. – Mason Oct 31 '22 at 02:53
  • $X$ its just normed linear vector space. i suppose that is an infinite dimensional vector space. The excercise don't say nothing else – LMOTRU Oct 31 '22 at 12:14
  • Hint:the basic open nbds of zero in the weak topology are of the form $\cap_{i=1}^{n} f_{i} ^{-1}(V_{i } }) $, for some open subsets $V_{i} $ of the real(or complex) numbers, in particular any such basic nbd, contains the intersection of the respective kernels, which is necessarly not trivial, since $X$ is infinite dimensional. – belkacem abderrahmane Oct 31 '22 at 12:23
  • @belkacemabderrahmane can you explain me the part of kernels? i've notice this solution in another question but i don't understand this – LMOTRU Oct 31 '22 at 19:58
  • the kernel of a Linear form has codimension one, the intersection of the kernels of two linear forms has codimension greater lower or equal two, u can prove by induction that the intersection of the kernels of n linear forms has codimension lower or equal n, in particular it has finite codimension in an infinite dimensional space, which implies in particular that it is not empty, so the basic nbd contains a line, this line meets, the unit sphere, i.e 0 belongs to the closure of the unit sphere – belkacem abderrahmane Oct 31 '22 at 20:19

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