# Pointwise convergence of continuous linear operators plus convergence of operator norms implies convergence in operator norm in Banach spaces?

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?

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 $$\|F_n-F\|= 2.$$