A more heterogeneous example of a vector space is data tracks. A depth-sounding survey of a lake can make a vector space that is a collection of tracks, a vector of vectors (each vector having a different number of components, because lakes are not square). This vector space of depths along tracks in a lake contains the depth values only. The (x,y)-coordinate information locating each measured depth value is (normally) something outside the vector space. A data space could also be a collection of echo soundings, waveforms recorded along tracks.
We briefly recall information about vector spaces found in elementary books: Let be any scalar. Then if is a vector and is conformable with it, then other vectors are and .The size measure of a vector is a positive value called a norm. The norm is usually defined to be the dot product (also called the L2 norm), say .For complex data it is where is the complex conjugate of .In theoretical work the ``length of a vector'' means the vector's norm. In computational work the ``length of a vector'' means the number of components in the vector.
Norms generally include a weighting function. In physics, the norm generally measures a conserved quantity like energy or momentum, so, for example, a weighting function for magnetic flux is permittivity. In data analysis, the proper choice of the weighting function is a practical statistical issue, discussed repeatedly throughout this book. The algebraic view of a weighting function is that it is a diagonal matrix with positive values spread along the diagonal, and it is denoted .With this weighting function the L2 norm of a data space is denoted .Standard notation for norms uses a double absolute value, where .A central concept with norms is the triangle inequality, whose proof you might recall (or reproduce with the use of dot products).