Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved. Let be a vector space, and its dual space. Consider, for example, an order-2 covariant tensor . Then can be identified with a bilinear form on . In other words, it is a function of two arguments in which can be represented as a pair of slots: Abstract index notation is merely a labelling of the slots with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical): A tensor contraction (or trace) between two tensors is represented by the repetition of an index label, where one label is contravariant (an upper index corresponding to the factor ) and one label is covariant (a lower index corresponding to the factor ). Thus, for instance, is the trace of a tensor over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or natural pairing) between tensor factors of type and those of type . Abstract indices and tensor spaces A general homogeneous tensor is an element of a tensor product of copies of and , such as Label each factor in this tensor product with a Latin letter in a raised position for each contravariant factor, and in a lowered position for each covariant position. In this way, write the product as or, simply The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example: Contraction In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated contraction (or trace) map. For instance, is the trace on