Finger vein recognition is an emerging biometric technology with high security and various application scenarios. Most finger vein recognition methods are based on a single view. However, the inherent problems in single-view finger vein recognition, such as limited feature, sensitivity to finger translation and rotation, and the ambiguity issue in 2D projections, hinder the improvement of the system performance. To address these problems and enhance finger vein verification performance, we employ multi-view finger vein images that are capable of providing a more comprehensive feature of 3D finger vein. Specifically, we design a novel low-cost full-view finger vein imaging device that enables full-view capture of finger veins with only a single camera and establish a multi-view finger vein dataset, named THU-MVFV. In addition, we propose a Multi-view Finger Vein Feature Encoding and Selection Network (MFV-FESNet), which is based on an improved Transformer encoder that can learn the dependencies between different views. By fusing the extracted global context feature and local dominant feature, the network can generate a feature descriptor with high discrimination. Extensive experiments are conducted on THU-MVFV and demonstrate the superior performance of the proposed model. The THU-MVFV dataset will be publicly available at https://github.com/Finger-Vein-Dataset/THU-MVFV.