By using an array antenna which has N receivers, How many targets could be detected and classified when we use the MUSIC(MUltiple SIgnal Classification) algorithm?
not a foolish question at all.
for a uniform linear array.
there are a number of different answers in the literature that do not agree.
the first issue is that direction finding and beamforming are different and have different upper bounds on the number of signals that can be resolved.
for beamforming $N-1$ sources under the narrow band assumption is typically the upper bound. you might assume polarization might double that number.
again under the narrow band assumption, the rank of the cross spectrum matrix is an upper bound for direction finding. One might reasonably assume $N$ for $N$ sensors but one can construct a virtual array from the negative coarray.
The virtual array is based subtracting outputs of pairs of real sensors. One can form $N^2$ such pairs if $x_k-x_j$ is distinct from $x_j-x_k$. If one were to believe this, the upper bound at a single frequency is $N^2$ sources.
The $N^2$ result is given in the peer reviewed literature by an eminent IEEE Fellow but I can’t say it is compatible with my intuition.