Sequence space

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequences spaces in analysis are the ℓ^p spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of L^p spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c₀, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

Sequence space (evolution)

In evolutionary biology, sequence space is a way of representing all possible sequences (for a protein, gene or genome). The sequence space has one dimension per amino acid or nucleotide in the sequence leading to highly dimensional spaces.

Most sequences in sequence space have no function, leaving relatively small regions that are populated by naturally occurring genes. Each protein sequence is adjacent to all other sequences that can be reached through a single mutation. Evolution can be visualised as the process of sampling nearby sequences in sequence space and moving to any with improved fitness over the current one.

Representation

A sequence space is usually laid out as a grid. For protein sequence spaces, each residue in the protein is represented by a dimension with 20 possible positions along that axis corresponding to the possible amino acids. Hence there are 400 possible dipeptides arranged in a 20x20 space but that expands to 10¹³⁰ for even a small protein of 100 amino acids arranges in a space with 100 dimensions. Although such overwhelming multidimensionality cannot be visualised or represented diagrammatically, it provides a useful abstract model to think about the range of proteins and evolution from one sequence to another.