= Solution
Work with real-valued functions and put $\operatorname{sign}(0)=0$. Let $r=f-p_*$, and write an arbitrary competitor as $p=p_*+v$ with $v\in U$. Pointwise,
$$
|r-v|\geq |r|-v\,\operatorname{sign}(r).
$$
For $r\ne0$ this is $|r-v|\geq\operatorname{sign}(r)(r-v)$; for $r=0$ the right side is zero. Integrating and using the assumed orthogonality gives
$$
\|f-p\|_1=\int|r-v|
\geq\int|r|-\int v\,\operatorname{sign}(r)
=\|f-p_*\|_1.
$$
Thus \b[$p_*$ is a best $L^1$ approximation from $U$]. This <sign criterion for best L1 approximation> is sufficient without any finite-dimensionality or uniqueness assumption; possible residual zeros do not invalidate the inequality.
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