does not define a norm, because it is not subadditive. On the other hand, the formula defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm, though, which is homogeneous of degree Hence, the function defines a metric. The metric space is denoted by Although the -unit ball around the origin in this metric is "concave", the topology defined on by the metric is the usual vector space topology of hence is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of is to denote by the smallest constant such that the scalar multiple of the -unit ball contains the convex hull of which is equal to The fact that for fixed we have shows that the infinite-dimensional sequence space defined below, is no longer locally convex. When There is one norm and another function called the "norm" (with quotation marks). The mathematical definition of the norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm which is discussed by Stefan Rolewicz in Metric Linear Spaces. The -normed space is studied in functional analysis, probability theory, and harmonic analysis. Another function was called the "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector Many authors abuse terminology by omitting the quotation marks. Defining the zero "norm" of is equal to This is not a norm because it is not homogeneous. For example, scaling the vector by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances. The -norm in infinite dimensions and spaces The sequence space The -norm can be extended to vectors that have an infinite number of components (sequences), which yields the space This contains as special cases: the space of sequences whose series is absolutely convergent, the space of square-summable sequences, which is a Hilbert space, and the space of bounded sequences. The space of sequences has a natural vector space structure by applying