Vector space is the high-dimensional mathematical space in which embedding vectors live, where each vector occupies a position defined by its coordinates and where geometric proximity corresponds to semantic similarity. It is the arena in which all vector search takes place.
The defining insight is that meaning maps to location. Because embeddings are constructed so that related content produces nearby vectors, the structure of the space reflects the structure of meaning: similar items cluster together, unrelated items lie far apart, and in some spaces directions correspond to meaningful transformations. This is what allows similarity to be computed as distance or angle rather than by symbolic comparison.
Reasoning about vector space requires some care, because high-dimensional spaces behave unlike the two- and three-dimensional spaces our intuition is built on — a phenomenon captured by the curse of dimensionality. Concepts like neighbourhoods and distance still apply, but their behaviour can be surprising. The terms vector space, embedding space, and latent space are largely interchangeable, all referring to this space where stored vectors reside and searches are performed.