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## Definition of a vector space

An operator transforms a space to another space. Examples of spaces are model space and data space . We think of these spaces as vectors with components packed with numbers, either real or complex numbers. The important practical concept is that not only does this packing include one-dimensional spaces like signals, two-dimensional spaces like images, 3-D movie cubes, and zero-dimensional spaces like a data mean, etc., but spaces can be mixed sets of 1-D, 2-D, and 3-D objects. One space that is a set of three cubes is the Earth's magnetic field, which has three components, each component being a function of three-dimensional physical space. (The 3-D physical space we live in is not the abstract vector space of models and data so abundant in this book. In this book the word space'' without an adjective means the  vector space.'') Other common spaces are physical space and Fourier space.

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 -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 norm), say . For complex data it is , where is the complex conjugate of . A notation that does transpose and complex conjugate at the same time is . In theoretical work, the size of a vector'' means the vector's norm. In computational work the size 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; therefore, 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 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, a proof you might recall (or reproduce with the use of dot products).