A multiresolution analysis (MRA) of multiplicity \(d\) of \(L^2({\mathbb{R}}^n)\) is a generalization of the usual MRA, where the sample spaces are generated by a refinable vector of \(d\) functions. The paper is concerned with the multivariate case, where the dilation matrix is \(2I\). In particular it can be shown that if the refinable function vector is compactly supported, then it is possible to find a compactly supported wavelet vector generating a semiorthogonal basis of \(L^2({\mathbb{R}}^n)\). Examples of semiorthogonal wavelets are given for continuous linear spline functions.