In the mathematical physics of quantum mechanics, **Liouville space**, also known as **line space**, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt inner product.^{[1]}^{[2]}

Abstractly, Liouville space is equivalent (isometrically isomorphic) to the tensor product of a Hilbert space with its dual.^{[1]}^{[3]} A common computational technique to organize computations in Liouville space is vectorization.^{[2]}

Liouville space underlies the density operator formalism and is a common computation technique in the study of open quantum systems.^{[2]}^{[3]}