A genome space is a moduli space of genomes. In this space, each point corresponds to a genome. The natural distance between two points in the genome space reflects the biological distance between these two genomes. Currently, there is no method to represent genomes by a point in a space without losing biological information. Here, we propose a new graphical representation for DNA sequences. The breakthrough of the subject is that we can construct the moment vectors from DNA sequences using this new graphical method and prove that the correspondence between moment vectors and DNA sequences is one-to-one. Using these moment vectors, we have constructed a novel genome space as a subspace in R(N). It allows us to show that the SARS-CoV is most closely related to a coronavirus from the palm civet not from a bird as initially suspected, and the newly discovered human coronavirus HCoV-HKU1 is more closely related to SARS than to any other known member of group 2 coronavirus. Furthermore, we reconstructed the phylogenetic tree for 34 lentiviruses (including human immunodeficiency virus) based on their whole genome sequences. Our genome space will provide a new powerful tool for analyzing the classification of genomes and their phylogenetic relationships.