On the non-uniqueness of minimal projection in LpLp spaces

The main objective of this note is to exhibit a simple example of subspaces U⊂Lp(μ)U⊂Lp(μ) (p≠2p≠2) that admit two different projections with minimal norm. While for p=1,∞p=1,∞, such subspaces are well-known [W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1