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Mar 7 at 3:24 vote accept mathex
Mar 6 at 7:14 comment added Martin Hairer @mathex No. It is densely defined linear functional that's unbounded, so cannot be extended to a continuous (and therefore bounded) linear functional.
Mar 6 at 0:35 comment added mathex Thank you Sir for your answer, for the second question, fixing $\omega \in \Omega,\overline{\xi}(\omega) \in\mathcal{E} '$ since $\overline{\mathcal{E}}=L^2$ (for the norm $\Vert\cdot \Vert_{L^2}$) then can't we extend $\overline{\xi}(\omega)$ to $L^2$?
Mar 5 at 19:29 history answered Martin Hairer CC BY-SA 4.0