Multi-view clustering and multi-view dimension reduction explore ubiquitous and complementary information between multiple features to enhance the clustering, recognition performance. However, multi-view clustering and multi-view dimension reduction are treated independently, ignoring the underlying correlations between them. In addition, previous methods mainly focus on using the tensor nuclear norm for low-rank representation to explore the high correlation of multi-view features, which often causes the estimation bias of the tensor rank. To overcome these limitations, we propose the partial tubal nuclear norm regularized multi-view learning (PTN2ML) method, in which the partial tubal nuclear norm as a non-convex surrogate of the tensor tubal multi-rank, only minimizes the partial sum of the smaller tubal singular values to preserve the low-rank property of the self-representation tensor. PTN2ML pursues the latent representation from the projection space rather than from the input space to reveal the structural consensus and suppress the disturbance of noisy data. The proposed method can be efficiently optimized by the alternating direction method of multipliers. Extensive experiments, including multi-view clustering and multi-view dimension reduction substantiate the superiority of the proposed methods beyond state-of-the-arts.