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In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus’s characterization of      A A      subspaces of      C [ a , b ] C[a,b]      . As a consequence we obtain that if the metric projection      P G {P_G}       from      C [ a , b ] C[a,b]       onto a finite-dimensional subspace      G G       has a continuous selection and elements of      G G       have no common zeros on      ( a , b ) (a,b)      , then      G G       is an      A A      -subspace.

Weak Chebyshev subspaces and 𝐴-subspaces of 𝐶[𝑎,𝑏]