Hyperspectral image classification is one of the most important techniques for analyzing hyperspectral image that have hundreds of spectrum luminance values of near-infrared to visible light. For this classification, supervised learning methods are widely used, but in general, they typically trade off between their accuracy and computational complexity. Our approach is based on a composite kernel method, and the computation is simplified to achieve higher processing speeds by efficiently using the hardware resources of FPGA. The accuracy of this approach reaches 98.0% and 98.8% on two benchmark datasets, Indian Pines and Salinas via simulation, which is comparable to those in previous works. Two implementations, one with less hardware resources but more off-chip memory bandwidth, and another with more hardware resources but less off-chip memory bandwidth, are implemented on an FPGA and evaluated. The processing speeds of the two implementations are the same, which is 1.3 Mpixels/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> for 2048 pixel wide images. This processing speed is fast enough for real-time processing, and faster than previous studies when normalized by hardware size and power consumption. We also introduce two more implementations that aim to reduce the on-chip memory usage of the second implementation within a reasonable increase of off-chip memory bandwidth, and we discuss which implementation is advantageous under what conditions.