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In the case of a regular grid with<jats:italic>n<\/jats:italic>grid points per direction, a spatial dimension<jats:italic>d<\/jats:italic>yields<jats:italic>n<jats:sup>d<\/jats:sup><\/jats:italic>grid points. A grid function defined on such a grid is an example of a tensor of order<jats:italic>d<\/jats:italic>. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly in<jats:italic>d<\/jats:italic>. In this way, data of size<jats:italic>n<jats:sup>d<\/jats:sup><\/jats:italic>= 1000<jats:sup>1000<\/jats:sup>can also be treated.<\/jats:p><jats:p>This paper introduces the algebraic and analytical aspects of tensor spaces. 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