Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual, this is enough to conclude the partial evaluation map is a linear isomorphism from M to M ∗ ∗ . This then allows the definition of the inverse partial evaluation map, η η − − 1 : M ∗ ∗ → → M , which allows the inverse metric to be defined as η η − − 1 : M ∗ ∗ × × M ∗ ∗ → → R , η η − − 1 ( α α , β β ) = η η ( η η − − 1 ( α α ) , η η − − 1 ( β β ) ) where the two different usages of η η − − 1 can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metric η is indeed the matrix inverse to η.