Alzheimer's Disease (AD), one of the main causes of death in elderly people, is characterize by Mild Cognitive Impairment (MCI) at prodromal stage. Nevertheless, only part of MCI subjects could progress to AD. The main objective of this paper is thus to identify those who will develop a dementia of AD type among MCI patients. 18 F-FluoroDeoxyGlucose Positron Emission Tomography (18 F-FDG PET) serves as a neuroimaging modality for early diagnosis as it can reflect neural activity via measuring glucose consumption at resting-state. In this paper, we design a deep network on 18 F-FDG PET modality to address the problem of AD identification at early MCI stage. To this end, a Multi-view Separable Pyramid Network (MiSePyNet) is proposed, in which representations are learned from axial, coronal and sagittal views of PET scans and then combined to make a decision jointly. Different from the widely and naturally used 3D convolution operations for 3D images, the proposed architecture is deployed with separable convolution from slicewise to spatial-wise successively, which can retain the spatial information and reduce training parameters compared to 2D and 3D networks, respectively. Experiments on ADNI dataset show that the proposed method is comparable to other state-of-the-art algorithms for classifying AD from Normal Control (NC). For predicting the progression of Mild Cognitive Impairment, our method can yield better performance than both traditional and deep learning-based algorithms, with a classification accuracy of 83.05%.