Based on the definition of local spectral subspace, we propose a novel approach called LOSP for local overlapping community detection. Instead of using the invariant subspace spanned by the dominant eigenvectors of the entire network, we run the power method for a few steps to approximate the leading eigenvectors that depict the embedding of the local neighborhood structure around seeds of interest. We then seek a sparse approximate indicator vector in the local spectral subspace spanned by these vectors such that the seeds are in its support. We evaluate LOSP on five large real world networks across various domains with labeled ground-truth communities and compare the results with the state-of-the-art community detection approaches. LOSP identifies the members of a target community with high accuracy from very few seed members, and outperforms the local Heat Kernel or PageRank diffusions as well as the global baselines. Two candidate definitions of the local spectral subspace are analyzed, and different community scoring functions for determining the community boundary, including two new metrics, are thoroughly evaluated. The structural properties of different seed sets and the impact of the seed set size are discussed. We observe low degree seeds behave better, and LOSP is robust even when started from a single random seed. Using LOSP as a subroutine and starting from each ego connected component, we try the harder yet significant task of identifying all communities a single vertex is in. Experiments show that the proposed method achieves high F1 measures on the detected multiple local overlapping communities containing the seed vertex.