Prove that a linear subspace of $C([0,1])$ is closed

functional-analysis vector-spaces uniform-convergence

The linear functional $L(x)=x(0)$ is a continuous linear map from $C[0,1]$ to $\mathbb{C}$ because it is bounded, i.e., $|T(x)|=|x(0)| \le \|x\|$. Therefore the inverse image of $\{0\}$ under $T$ is closed, which is the subspace $W$.

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