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For a normed vector space $(V,\|\cdot\|_{V})$, let $A\subseteq V$ be compact, and let $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$ be the space of continuous functions from $A$ to $\mathbb{R}$ with respect to the sup-norm. How can I use specifically the Weierstrass M-Test, and facts about absolutely convergent series in normed vector spaces to show that $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$ is a Banach Space?
Can I assume that I have an arbitrary absolutely convergent series $\Bigl(\sum_{k=0}^n f_{k}\Bigr)_{n\in\mathbb{N}}$ in $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$, and then just let $M_n=\|f_n\|_{\infty}$ and then apply the M-Test to show that the sequence converges?

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    $\begingroup$ Yes, that's the idea. Absolute convergence implying convergence is equivalent to completeness in normed linear spaces, see here: math.stackexchange.com/questions/2180369/… $\endgroup$ – Theo Bendit Apr 4 at 0:21
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    $\begingroup$ If a sequence of continuous functions $(F_n)$ converges uniformly then the limit $F$ is continuous, this is a well-known exercice $|F(x)-F(a)|\le 2\|F-F_n\|_\infty + |F_n(x)-F_n(a)|$. For $n$ large enough $\|F-F_n\|_\infty < \epsilon/3$ and for $x$ close enough to $a$, $|F_n(x)-F_n(a)| < \epsilon/3$ $\endgroup$ – reuns Apr 4 at 0:24

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