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Tensors in brief tensors Definition Given a vector space $V$ and its dual space $V^\text{}$ , a rank $\left( p, q \right)$ tensor is a multilinear map $\pmb{T} \in \underset{i=1}{\overset{p}{\bigotimes}} V \underset{j=1}{\overset{q}{\bigotimes}} V^\text{}$ .
Tensor networks tensor-networks tensors contractions graphs gauge-invariance Motivation When we use the traditional index notation for manipulating tensors, we are essentially keeping track of the following information: The invariant objects concerned, or tensors, which are labelled using letters.
Tensor fields tensors Tuple index notation To make the tensor notation a little less messy, let us use the tuple index notation, where tuples of indices are replaced by their capital letter.