For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a non-unique function $h\in K$ which minimizes $||f-g||_p ,g\in K.$ I'd appreciate some help finding such examples.
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For $L^1$, take $K$ to be the set of functions for which $\int h \mathrm{d}x = 0$. Let $f$ be the characteristic function of the interval $[0,1]$. Note $$ 1 = \int h - f \mathrm{d}x \leq \|h - f\|_1 $$ Note that this minimum is achieved for any $h$ of the form: $h = +1$ on a measure 1/2 subset of $[0,1]$, $h = -1$ on its complement in $[0,1]$, and $h = 0$ otherwise.
For $L^\infty$ take $K$ to be the set of essentially bounded functions vanishing on $[0,1]$. Take $f \equiv 1$.
