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$\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac measuredelta distribution supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.
$\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac measure supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.
$\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac delta distribution supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.
$\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac measure supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.