It is known (Riesz Theorem) that every linear functional $f$ on $X=C[a,b]$ can be represented as $\int_a^bx(t)dv(t)$ for all $x\in X$, where $v$ has bounded variation. Is it true that every linear functional $f$ on $Y=C^1[a,b]$ can be represented as $\int_a^by(t)v(t)dt$ for some continuous $v$?
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1How about $$f \mapsto \int_0^{\frac12} f(t),dt$$ on the space $C^1[0,1]$? – mechanodroid Jul 04 '22 at 13:53