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Vector space of infinite sequences
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
Sequence_space
Finite or infinite ordered list of elements
studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for
Sequence
Type of vector space in math
Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized
Hilbert_space
Representation of possible genetic sequences
evolutionary biology, sequence space is a way of representing all possible sequences (for a protein, gene or genome). The sequence space has one dimension
Sequence_space_(evolution)
Function spaces generalizing finite-dimensional p norm spaces
{\displaystyle \ell ^{\infty },} the space of bounded sequences. The space of sequences has a natural vector space structure by applying scalar addition
Lp_space
In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below
Orlicz_sequence_space
Normed vector space that is complete
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
Banach_space
Space of bounded sequences
{\displaystyle \ell ^{\infty }} , the (real or complex) vector space of bounded sequences with the supremum norm, and L ∞ = L ∞ ( X , Σ , μ ) {\displaystyle
L-infinity
Rational design of new protein molecules
designed anew, based on no prior sequence. Both de novo designs and protein redesigns can establish rules on the sequence space: the specific amino acids that
Protein_design
Mathematical operator acting on sequences
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear
Sequence_transformation
Function space
normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequence spaces d (
Lorentz_space
Sequence of points that get progressively closer to each other
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given
Cauchy_sequence
Mathematical function whose set of values is bounded
natural number n {\displaystyle n} . The set of all bounded sequences forms the sequence space l ∞ {\displaystyle l^{\infty }} .[citation needed] The definition
Bounded_function
Type of mathematical space
is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value
Compact_space
Set of genes all related by point mutations that have equivalent function or fitness
can randomly move through neutral networks and traverse regions of sequence space which may have consequences for robustness and evolvability. Neutral
Neutral_network_(evolution)
Neurological condition involving the crossing of senses
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
Synesthesia
Metric geometry
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
Complete_metric_space
Space of bounded sequences
mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences ( x n ) {\displaystyle \left(x_{n}\right)}
C_space
Abstract strategy board and card game
and a free corner space) is counted as a sequence. Sequences of the same color may intersect, but only at a single position. Sequence rules dictate no
Sequence_(game)
Value to which an infinite sequence tends
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
Limit_of_a_sequence
number arithmetic" for the purposes of manipulating and comparing these sequence numbers. In short, when the absolute serial number value decreases by more
Serial_number_arithmetic
Generalization of a sequence of points
or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize
Net_(mathematics)
Protein engineering method
generation of a library of variant genes. The sequence space for random sequence is vast (10130 possible sequences for a 100 amino acid protein) and extremely
Directed_evolution
Science fiction series by Stephen Baxter
The Xeelee Sequence (/ˈziːliː/; ZEE-lee) is a series of hard science fiction novels, novellas, and short stories written by British author Stephen Baxter
Xeelee_Sequence
Measure of the "size" of linear operators
infinite-dimensional example, consider the sequence space ℓ 2 , {\displaystyle \ell ^{2},} which is an Lp space, defined by ℓ 2 = { ( a n ) n ≥ 1 : a n ∈
Operator_norm
Topological space characterized by sequences
space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that
Sequential_space
Type of numeric sequence used in statistics
Halton sequences are sequences used to generate points in space for numerical methods such as Monte Carlo simulations. Although these sequences are deterministic
Halton_sequence
(more resolvable) problems and conditions on sequence spaces. Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured
Commutator_subspace
Infinite sequence of numbers satisfying a linear equation
linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence. For example, the Fibonacci sequence 0 , 1 , 1 , 2
Constant-recursive_sequence
Algebraic tool for computing topological spaces' invariants
homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces. The result is due to two Austrian
Mayer–Vietoris_sequence
Type of convergence in Hilbert spaces
Hilbert space is the convergence of a sequence of points in the weak topology. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Hilbert space H {\displaystyle
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Series of urban open spaces and pedestrian zones in Portland, Oregon, U.S.
The Halprin Open Space Sequence is a series of urban open spaces and pedestrian zones between Southwest Lincoln Street and Clay Street, in downtown Portland
Halprin_Open_Space_Sequence
Sequence space that is Banach
mathematics, a BK-space or Banach coordinate space is a sequence space endowed with a suitable norm to turn it into a Banach space. All BK-spaces are normable
BK-space
Space Shuttle orbiter (1985–2011)
(OV-104) 16 April 2007: Consolidated Launch Manifest: Space Shuttle Flights and ISS Assembly Sequence Space Shuttle Atlantis: Last Flight – slideshow by Life
Space_Shuttle_Atlantis
Mathematical metric
particular, in optimization algorithms for these. For the sequence space of infinite-length sequences of real or complex numbers, the Chebyshev distance generalizes
Chebyshev_distance
Mathematical set with some added structure
Quadratic space Quotient space (disambiguation) Riemann's Moduli space Sample space Sequence space Sierpiński space Sobolev space Standard space State space Stone
Space_(mathematics)
Short story by Jorge Luis Borges
said with two characters." The full possible set of protein sequences (protein sequence space) has been compared to the Library of Babel. In the Library
The_Library_of_Babel
Short exact sequence of sheaves on projective space
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of
Euler_sequence
Infinite sequence in mathematics
Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths
Kolakoski_sequence
Concept in set theory
the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space is commonly
Baire_space_(set_theory)
Set of functions between two fixed sets
spaces and Banach spaces. In functional analysis, the set of all functions from the natural numbers to some set X is called a sequence space. It consists of
Function_space
takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups. A sequence of pointed
Puppe_sequence
Mathematical concept
property. The spaces ℓ p {\displaystyle \ell ^{p}} for p ≠ 2 {\displaystyle p\neq 2} and c 0 {\displaystyle c_{0}} (see Sequence space) have closed subspaces
Approximation_property
is a certain linear subspace of the algebraic dual of a sequence space. Given a sequence space X, the β-dual of X is defined as X β := { x ∈ K N :
Beta-dual_space
1968 film by Stanley Kubrick
long sequences are accompanied only by music. Shunning the convention that major film productions should feature original music, 2001: A Space Odyssey
2001:_A_Space_Odyssey
Continuous band of stars that appears on plots of stellar color versus brightness
In astronomy, the main sequence is a classification of stars which appear on plots of stellar color versus brightness as a continuous and distinctive band
Main_sequence
Mathematical inequality relating inner products and norms
finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums
Cauchy–Schwarz_inequality
Sequence space that is Fréchet
a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with
FK-space
Spectral sequence in algebraic topology
Serre fibration of topological spaces, and let F be the (path-connected) fiber. The Serre cohomology spectral sequence is the following: E 2 p , q = H
Serre_spectral_sequence
Topological space where every sequence has a convergent subsequence
In mathematics, a topological space X {\displaystyle X} is sequentially compact if every sequence of points in X {\displaystyle X} has a convergent subsequence
Sequentially_compact_space
Method used for display options on video text terminals
ANSI escape sequences are a standard for in-band signaling to control cursor location, color, font styling, and other options on video text terminals
ANSI_escape_code
Value approached by a mathematical object
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are
Limit_(mathematics)
Compact operator for which a finite trace can be defined
noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space ℓ 1 ( N ) . {\displaystyle
Trace_class
Concept in mathematical topology
Hausdorff topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some
Hemicompact_space
Concept in cheminformatics
chemical space. Cheminformatics Drug design Sequence space (evolution) Molecule mining Reymond, J.-L.; Awale, M. (2012). "Exploring chemical space for drug
Chemical_space
Long exact sequence
topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle
Gysin_homomorphism
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displaystyle \mathbb
Bolzano–Weierstrass_theorem
Type of sequence in numerical analysis
Sobol' sequences (also called LPτ sequences or (t, s) sequences in base 2) are a type of quasi-random low-discrepancy sequence. They were first introduced
Sobol_sequence
Population structure of viruses with a large number of variant genomes
of dominance of the master sequence and drift of the population in sequence space. The core quasispecies concepts are described by two fundamental equations:
Viral_quasispecies
Cluster point in a topological space
for sequences. A cluster point or accumulation point of a sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} in a topological space X {\displaystyle
Accumulation_point
the ARPANET sequence bug, and OSPF version 2 replaced it with a linear numbering space, with special rules for what happens when the sequence numbers reach
Lollipop_sequence_numbering
Type of function space
Orlicz spaces are many of the most important Sobolev spaces. In addition, the Orlicz sequence spaces are examples of Orlicz spaces. These spaces are called
Orlicz_space
Tool in homological algebra
spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and
Spectral_sequence
FK-AK spaces are separable spaces. BK-space – Sequence space that is Banach FK-space – Sequence space that is Fréchet Normed space – Vector space on which
FK-AK_space
mathematics, a sequence of vectors (xn) in a Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,\langle \cdot ,\cdot \rangle )} is called a Riesz sequence if there
Riesz_sequence
Topological space with a dense countable subset
mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle
Separable_space
Process in bioinformatics that identifies equivalent sites within molecular sequences
In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence
Sequence_alignment
Stellar classification
A K-type main-sequence star (also called a "K-type dwarf" or "orange dwarf") is a main-sequence (core hydrogen-burning) star of spectral type K. The spectral
K-type_main-sequence_star
Deep learning architecture
processing long sequences, and it is based on the Structured State Space sequence (S4) model. To enable handling long data sequences, Mamba incorporates
Mamba (deep learning architecture)
Mamba_(deep_learning_architecture)
Bounds of a sequence
inferior and limes superior) of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a
Limit inferior and limit superior
Limit_inferior_and_limit_superior
Stellar classification
An F-type main-sequence star is a main-sequence, core-hydrogen-fusing star of spectral type F. The spectral luminosity class is V. They have from around
F-type_main-sequence_star
is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to OEIS
List_of_integer_sequences
Binary sequence used in telecommunications
Hadamard code Hamming code JPL code Kasami code Zadoff–Chu sequence Complementary sequences Space Network – a NASA system that uses Gold codes George, Maria;
Gold_code
DNA oligonucleotides that can perform a specific chemical reaction
sequence space cannot be surveyed. However, since there are likely many potential candidates for a given catalytic reaction within the sequence space
Deoxyribozyme
Type of continuous linear operator
such as a matrix. In infinite-dimensional spaces, bounded sets are usually not compact, and bounded sequences need not have convergent subsequences. Compact
Compact_operator
Operator on a Hilbert space that shifts basis vectors
shift operator, that is, a shift operator acting on one-sided sequences or shift spaces. The term "operator" is used to draw contrast to finite-dimensional
Unilateral_shift_operator
Vector space with generalized dot product
product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If
Inner_product_space
First known wavelet basis
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is
Haar_wavelet
separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented
Calkin_correspondence
Type of topological space
topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and
Hausdorff_space
Linear subspace generated from a vector acted on by a power series of a matrix
{\displaystyle (k^{n})^{A}} , where k n {\displaystyle k^{n}} is the linear space on k {\displaystyle k} . k n {\displaystyle k^{n}} can be decomposed as
Krylov_subspace
Kind of linear transformation
finite-dimensional normed space is bounded. On the sequence space c 00 {\displaystyle c_{00}} of eventually zero sequences of real numbers, considered
Bounded_operator
Topological space where each point has a countable neighbourhood basis
sequence in A {\displaystyle A} converging to x . {\displaystyle x.} A space with this sequence property is sometimes called a Fréchet–Urysohn space.
First-countable_space
Computer text file character representing blank space
but displayed as an empty space, like a Hangul block containing no jamo. It is used in KS X 1001 Hangul combining sequences to introduce them or denote
Whitespace_character
Biomolecule consisting of chains of amino acid residues
of changes in DNA and RNA over time Protein sequence space – Representation of possible genetic sequences Protein superfamily – Grouping of proteins Protein
Protein
Modular space station in low Earth orbit
Retrieved 29 January 2020. "Axiom Station Assembly Sequence – Axiom Space Axiom Space". Axiom Space. Archived from the original on 10 August 2023. Retrieved
International_Space_Station
Topological vector spaces
Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Noncommutative geometric structure
a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators
Singular_trace
Surjective bounded operator on a Hilbert space preserving the inner product
real. They are the unitary operators on Rn. The bilateral shift on the sequence space ℓ2 indexed by the integers is unitary. The unilateral shift (right shift)
Unitary_operator
provides method to find them Complete metric space Topological space Function space Sequence space Compact space Lebesgue measure Outer measure Hausdorff
List_of_real_analysis_topics
On when a family of real, continuous functions has a uniformly convergent subsequence
generated Hausdorff space into a uniform space to be compact in the compact-open topology; see Kelley (1991, page 234). By definition, a sequence { f n } n ∈
Arzelà–Ascoli_theorem
Computational tool
vector space over the field F. A Schauder basis is a sequence {bn} of elements of V such that for every element v ∈ V there exists a unique sequence {αn}
Schauder_basis
Sequence of equally spaced numbers
An arithmetic progression, arithmetic sequence or linear sequence is a sequence of numbers such that the difference from any succeeding term to its preceding
Arithmetic_progression
Mode of convergence of a function sequence
{\displaystyle E} is a topological space, M {\displaystyle M} is a metric space, and ( f n ) {\displaystyle (f_{n})} is a sequence of continuous functions f n
Uniform_convergence
German chemist and protein engineer (born 1969)
exchanges and developed strategies to efficiently explore the protein sequence space (e.g. KnowVolution) and discovered protein engineering principles. Ulrich
Ulrich_Schwaneberg
Sequence of amino acids
leader sequence is a sequence of amino acids which, when attached to a protein, directs the protein to the bacterial periplasm, where the sequence is removed
PelB_leader_sequence
Universal C*-algebra
C*-algebra, meaning that as a Hilbert space, O n {\displaystyle {\mathcal {O}}_{n}} is isometric to the sequence space l 2 ( N ) {\displaystyle l^{2}(\mathbb
Cuntz_algebra
Intergenic DNA sequence separating ribosomal RNA genes
(1983-05-25). "Structure of mouse rRNA precursors. Complete sequence and potential folding of the spacer regions between 18S and 28S rRNA". Nucleic Acids Research
Internal_transcribed_spacer
Storytelling technique
designed sharply angled sets inspired by his own dream space. Ingmar Bergman lit dream sequences in several films with a harsh glare of light which he
Dream_sequence
Mathematical space with a notion of closeness
space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces,
Topological_space
SEQUENCE SPACE
SEQUENCE SPACE
Girl/Female
Tamil
Anuloma | அநà¯à®²à¯‹à®®à®¾
Sequence
Anuloma | அநà¯à®²à¯‹à®®à®¾
Biblical
a dividing; a sentence
Girl/Female
Hindu, Indian
Sentence
Boy/Male
Tamil
Symbol, First word in a sentence
Boy/Male
Indian, Sikh
Music; In-sequence
Girl/Female
Biblical
Dividing, sentence.
Girl/Female
Biblical
A dividing, a sentence.
Girl/Female
Tamil
Line, Sentence
Girl/Female
Indian, Sanskrit
Sentence
Boy/Male
Hindu
Sentence
Girl/Female
Indian
Sentence, Writing, Essay
Biblical
dividing, sentence
Boy/Male
Tamil
Symbol, First word in a sentence
Boy/Male
Indian, Sanskrit
Order; Sequence
Girl/Female
Hindu
Line, Sentence
Girl/Female
Muslim
Sentence, Writing, Essay
Girl/Female
Hindu
Line, Sentence
Girl/Female
Tamil
Line, Sentence
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Sequence
Boy/Male
Tamil
Sentence
SEQUENCE SPACE
SEQUENCE SPACE
Girl/Female
Indian, Punjabi, Sikh
Mind Full of Love
Male
Italian
Italian form of Latin Hercules, ERCOLE means "glory of Hera."
Biblical
flower; branch; a lock of hair
Girl/Female
Hindu
Star
Boy/Male
Arabic, Muslim
Colour of Allah
Girl/Female
American, British, English, French
Brilliant; Bright
Girl/Female
Muslim
Full of life
Surname or Lastname
English
English : habitational name for someone from a place called Vint, probably one in Oath, Somerset.Scottish : variant of Wint, which is of unexplained origin.
Boy/Male
Muslim/Islamic
Rising Standing, Existing, well-grounded
Girl/Female
Latin American English
Laurel tree or sweet bay tree (symbols of honour and victory).
SEQUENCE SPACE
SEQUENCE SPACE
SEQUENCE SPACE
SEQUENCE SPACE
SEQUENCE SPACE
pl.
of Sequela
n.
A melodic phrase or passage successively repeated one tone higher; a rosalia.
n.
A hymn introduced in the Mass on certain festival days, and recited or sung immediately before the gospel, and after the gradual or introit, whence the name.
v. t.
To utter sententiously.
n.
That which follows or succeeds as an effect; sequel; consequence; result.
n.
All five cards, of a hand, in consecutive order as to value, but not necessarily of the same suit; when of one suit, it is called a sequence flush.
n.
A want of grammatical sequence or coherence in a sentence; an instance of a change of construction in a sentence so that the latter part does not syntactically correspond with the first part.
adv.
In natural sequence; consequently; so.
n.
Simple succession, or the coming after in time, without asserting or implying causative energy; as, the reactions of chemical agents may be conceived as merely invariable sequences.
imp. & p. p.
of Sentence
n.
Three or more cards of the same suit in immediately consecutive order of value; as, ace, king, and queen; or knave, ten, nine, and eight.
n.
That which follows as a result; a sequence.
n.
The state of being sequent; succession; order of following; arrangement.
n.
A combination of words which is complete as expressing a thought, and in writing is marked at the close by a period, or full point. See Proposition, 4.
p. pr. & vb. n.
of Sentence
v. t.
To pass or pronounce judgment upon; to doom; to condemn to punishment; to prescribe the punishment of.
n.
Any succession of chords (or harmonic phrase) rising or falling by the regular diatonic degrees in the same scale; a succession of similar harmonic steps.
n.
A going before; anticipation in sequence or order.
a.
Lacking grammatical sequence.
v. t.
To decree or announce as a sentence.