The large number of bands contained in hyperspectral images (HSIs) can provide a wealth of information on regions of interest but also causes the "curse of dimensionality" and information redundancy. To address this issue, this paper proposes a novel band selection method based on triple constraints and attention network (TCANet-BS) for HSIs. The proposed TCANet-BS can make good use of band representativeness, band informativeness, and inter-band correlation when selecting bands. Specifically, TCANet-BS calculates band representativeness by taking advantage of the attention reconstruction network. Moreover, TCANet-BS obtains band informativeness and inter-band correlation through spectral information divergence and orthogonal subspace projection technique, respectively. Experimental results on the real-world hyperspectral dataset show that TCANet-BS can effectively improve classification accuracy compared with other advanced band selection methods.