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Let $X$ be a complex Banach space and $T: X → X$ be a bounded linear operator.

In the rest of this question, let $X=L^{∞}((0,1))$ be the space of complex-valued essentially bounded functions on $(0,1)$, equipped with its standard norm, and view $C([0,1])$ as a subspace of $X$.

Let $x_0 ∈[0,1]$.

Let $e_0, e_1 ∈ C([0,1])$ be two continuous functions such that $e_0(0)=e_1(1)=1$ and $e_0(1)=e_1(0)=0$. Let $T: X → X$ be given by \[ T f(x)=ℓ_0(f) e_0(x)+ℓ_1(f) e_1(x), \] where $ℓ_0$ and $ℓ_1$ are as defined in part (b).