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I want to calculate the dual space of $\mathcal{C}_0[a,b]$, that is the space of continuous functions on $[a,b]$ vanishing at $a$. I know that the dual of $\mathcal{C}[a,b]$ is the space of differences of Lebesgue-Stieltjes measures associated to increasing and left-continuous functions, vanishing at a point of $[a,b]$. Anyone can help me?

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Consider functional $$ \mathrm{ev}_a:\mathcal{C}([a,b])\to \mathbb{C}:f\mapsto f(a) $$ Note that for any closed subspace $E$ of Banach space $X$ holds $$ E^*=X^*/E^\perp $$ where $$ E^\perp:=\{f\in X^*:f(E)=\{0\}\} $$ Apply this to $E=\mathrm{Ker}(\mathrm{ev}_a)=\mathcal{C}_0([a,b])$, and $X=\mathcal{C}([a,b])$ to get $$ \mathcal{C}_0([a,b])^*=\mathcal{C}([a,b])^* / \mathrm{Ker}(\mathrm{ev}_a)^\perp =\mathcal{M}([a,b]) / \mathrm{span}\{\delta_a\} $$ where $\delta_a$ is a Dirac delta measure centered at point $a$.

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