31. Orthogonality is viewed as desirable by some researchers because it minimizes redundancy between the dimensions.32. Using a lattice basis, neither orthogonality nor unit length is guaranteed, only independence. 33. Under this new regime orthogonality became undesirable. 34. Preserving orthogonality requires that it is a rotation that leaves the sub-space invariant. 35. They satisfy well-defined orthogonality relations. 36. The orthogonality ensures that they do not interfere with each other and can be sent simultaneously. 37. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. 38. Not all projections that preserve the orthogonality of parallels and meridians are angle-preserving .) 39. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system.40. The author applies these results to generalized orthogonality relations and thence to Riesz and Hilbert spaces.
