 It is worth noting that neither \(\Phi\) nor \(\Phi^x\) is a Hilbert space in that each lacks an inner product that induces a metric with respect to which the space is complete, though for each space there is a topology with respect to which the space is complete. Nevertheless, each of them is closely related to the Hilbert space \(\Eta\) from which they are generated: \(\Phi\) is densely embedded in \(\Eta\), which in turn is densely embedded in \(\Phi^x\). Two other points are worth noting. First, dual pairs of this sort can also be generated from a pre-Hilbert space, which is a space that has all the features of a Hilbert space except that it is not complete, and doing so has the distinct advantage of avoiding the partitioning of functions into equivalence classes (in the case of functions spaces). The term rigged Hilbert space is typically used broadly to include dual pairs generated from either a Hilbert space or a pre-Hilbert space. Second, the term Gelfand triplet is sometimes used instead of the term rigged Hilbert space, though it refers to the ordered set \((\Phi , \Eta , \Phi^x)\), where \(\Eta\) is the Hilbert space used to generate \(\Phi\) and \(\Phi^x\).