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Let $K$ be a compact space and consider $\mathcal{C}(K)$ the space of continuous functions with the $\sup$ norm $\lVert f\rVert_\infty=\sup_{x\in K}|f(x)|$. Let also $\mathcal{L}:=\mathcal{L}(\mathcal{C}(K),\mathcal{C}(K))$ the set of continuous linear operators $\mathcal{C}(K)\to \mathcal{C}(K)$ with the operator norm.

If a sequence $F_n$ in $\mathcal{L}$ converges pointwisely to $F\in\mathcal{L}$ and $\lim_n\lVert F_n\rVert_{\operatorname{op}}=\lVert F\rVert_{\operatorname{op}}$, then $\lim_n\lVert F_n-F\rVert_{\operatorname{op}}=0$?

I know that this would be true if we were dealing with a Hilbert space, but does it still hold true in this Banach space?

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For $K=[0,1]$ let $$F_nf={1\over n}\sum_{k=1}^nf(k/n)\,{\bf 1}$$ Then $$F_nf\to Ff:=\int\limits_0^1f(t)\,dt\, {\bf 1}$$ Moreover $\|F_n\|=\|F\|=1.$ However $\|F_n-F\|\ge 1.$ Actually $\|F_n-F\|= 2.$

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