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We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\\mathbb{R}^\\Gamma$ for some $\\Gamma$. Consequently, a Fr\\'{e}chet space $E$ is weakly Ascoli iff $E=\\mathbb{R}^N$ for some $N\\leq\\omega$. 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