- #1
DavideGenoa
- 155
- 5
I read that in any locally convex topological space [itex]X[/itex], not necessarily a Hausdorff space but with linear operations continuous, for any ##x_0\ne 0## we can define a continuous linear functional [itex]f:X\to K[/itex] such that [itex]f(x_0)\ne 0[/itex].
I cannot find a proof of that anywhere and cannot prove it myself. Please correct me if I am wrong, but I think that if [itex]A[/itex] is closed in [itex]X[/itex] and [itex]x_0\in A[/itex] there exist a continuous linear functional rigorously separating [itex]x_0[/itex] and [itex]A[/itex], but I am not sure whether we can use that...
I have also tried with Minkowski functional, but with no result...
[itex]\infty[/itex] thanks for any help!
I cannot find a proof of that anywhere and cannot prove it myself. Please correct me if I am wrong, but I think that if [itex]A[/itex] is closed in [itex]X[/itex] and [itex]x_0\in A[/itex] there exist a continuous linear functional rigorously separating [itex]x_0[/itex] and [itex]A[/itex], but I am not sure whether we can use that...
I have also tried with Minkowski functional, but with no result...
[itex]\infty[/itex] thanks for any help!
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