For usage in evolutionary biology, see Sequence space (evolution). For mathematical operations on sequence numbers, see Serial number arithmetic.
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 sequence spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp 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.
mathematics, a sequencespace is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements...
studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for...
Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized...
{\displaystyle \ell ^{\infty },} the space of bounded sequences. The space of sequences has a natural vector space structure by applying addition and scalar...
In mathematics, an Orlicz sequencespace is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below...
the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish...
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given...
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequencespace). Sequence transformations include linear...
metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete...
natural number n {\displaystyle n} . The set of all bounded sequences forms the sequencespace l ∞ {\displaystyle l^{\infty }} .[citation needed] The definition...
normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequencespaces d (...
if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass...
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the lim {\displaystyle \lim } symbol...
number arithmetic" for the purposes of manipulating and comparing these sequence numbers. In short, when the absolute serial number value decreases by more...
Protein primary structure is the linear sequence of amino acids in a peptide or protein. By convention, the primary structure of a protein is reported...
The Xeelee Sequence (/ˈziːliː/; ZEE-lee)[a] is a series of hard science fiction novels, novellas, and short stories written by British science fiction...
spaces and Banach spaces. In functional analysis, the set of all functions from the natural numbers to some set X is called a sequencespace. It consists of...
Halton sequences are sequences used to generate points in space for numerical methods such as Monte Carlo simulations. Although these sequences are deterministic...
colored. In spatial-sequence, or number form synesthesia, numbers, months of the year, or days of the week elicit precise locations in space (e.g., 1980 may...
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X...
infinite-dimensional example, consider the sequencespace ℓ 2 , {\displaystyle \ell ^{2},} which is an Lp space, defined by ℓ 2 = { ( a n ) n ≥ 1 : a n ∈...
(OV-104) 16 April 2007: Consolidated Launch Manifest: Space Shuttle Flights and ISS Assembly SequenceSpace Shuttle Atlantis: Last Flight – slideshow by Life...
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