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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Sequences</title></head> <body> <h1 id="sequences">Sequences</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <li><a href='#notation'>Notation</a></li> <li><a href='#Generalization'>Generalisations</a></li> <ul> <li><a href='#nets'>Nets</a></li> <li><a href='#sequential_nets'>Sequential nets</a></li> </ul> <li><a href='#sequence_types'>Sequence types</a></li> <li><a href='#sequence_spaces'>Sequence spaces</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>A <strong>sequence</strong> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> whose <a class="existingWikiWord" href="/nlab/show/source">domain</a> is a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> (or more generally a subset of the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>; cf. <em><a class="existingWikiWord" href="/nlab/show/bi-infinite+sequence">bi-infinite sequence</a></em> and the further generalizations <a href="#Generalizations">below</a>). Often one means an <strong>infinite sequence</strong>, which is a sequence whose domain is infinite. Sequences (especially finite ones) are often called <strong><a class="existingWikiWord" href="/nlab/show/lists">lists</a></strong> in computer science. (In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, the domain of a sequence must be a <a class="existingWikiWord" href="/nlab/show/decidable+subset">decidable subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>.)</p> <p>Sequences may also be indicated by functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\omega \to X</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is the first countably infinite <a class="existingWikiWord" href="/nlab/show/ordinal">ordinal</a>: the key piece of structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> relevant for the study of sequences, particularly in analysis, is the order structure.</p> <p>Up to <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>, the only possible <a class="existingWikiWord" href="/nlab/show/domains">domains</a> are those of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>ℤ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{i\colon \mathbb{Z} \;|\; 0 \leq i \lt n\} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n = 0, 1, 2, \ldots, \infty</annotation></semantics></math>; other domains are used for notational convenience.</p> <p>An alternative generalisation takes the domain to be a set of <a class="existingWikiWord" href="/nlab/show/ordinal+numbers">ordinal numbers</a>, without loss of generality the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mo lspace="0em" rspace="thinmathspace">Ord</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mo><</mo><mi>α</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{i\colon \Ord \;|\; i \lt \alpha\} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> some specific ordinal number (or the <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Ord</mo></mrow><annotation encoding="application/x-tex">\Ord</annotation></semantics></math> of all ordinal numbers, if one wishes to allow for a proper class).</p> <p>A <em>subsequence</em> of a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><msub><mi>a</mi> <mi>n</mi></msub><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a = a_n: \mathbb{N} \to X</annotation></semantics></math> is a composition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mover><mo>→</mo><mi>i</mi></mover><mi>ℕ</mi><mover><mo>→</mo><mi>a</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \stackrel{i}{\to} \mathbb{N} \stackrel{a}{\to} X</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is an order-preserving monomorphism. The salient point is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/cofinal+diagram">cofinal</a> as an embedding.</p> <h2 id="notation">Notation</h2> <p>One normally writes the value of the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> at the argument <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a(i)</annotation></semantics></math>. Similarly, given a term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[i]</annotation></semantics></math> with the free variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, one often defines a sequence whose values equal those terms as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo><</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(a[i])_{i \lt n}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\{a[i]\}_i</annotation></semantics></math>, or the like. In fact, one even often says literally ‘Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_i)</annotation></semantics></math> be a sequence.’ even though ‘Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> be a sequence.’ would be less of an abuse of notation. This is all because notation for sequences arose before <a class="existingWikiWord" href="/nlab/show/functions">functions</a> were considered in their full generality, and one distinguished a ‘sequence’ (whose domain was a set of integers) from a ‘function’ (whose domain was an interval in the real line or a region in the complex plane). Early mathematicians also often conflated the sequence (the function itself) with its range (a subset of the function's <a class="existingWikiWord" href="/nlab/show/target">target</a>); hence the use of curly braces. All of this applies in greater generality to <a class="existingWikiWord" href="/nlab/show/families">families</a> with index sets other than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>.</p> <h2 id="Generalization">Generalisations</h2> <h3 id="nets">Nets</h3> <p>Infinite sequences are often used in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, but for topology in general, one needs to generalise to <a class="existingWikiWord" href="/nlab/show/nets">nets</a>, also called <em>Moore–Smith sequences</em>. Here one replaces the domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> by any arbitrary <a class="existingWikiWord" href="/nlab/show/direction">directed set</a>.</p> <h3 id="sequential_nets">Sequential nets</h3> <p>Recall that <a class="existingWikiWord" href="/nlab/show/weak+countable+choice">weak countable choice</a> is a rather weak version of the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> that is accepted even in most schools of <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>; it follows separately from both <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> and <a class="existingWikiWord" href="/nlab/show/countable+choice">countable choice</a>. However, when it fails (as it does in the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of some widely studied <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a>, such as the <a class="existingWikiWord" href="/nlab/show/topos+of+sheaves">topos of sheaves</a> over the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>), then some important results about sequences fail, including many standard results in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>. In this case, we may want a slight generalisation that we call <em>sequential nets</em>.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/sequential+net">sequential net</a></strong> is a <a class="existingWikiWord" href="/nlab/show/multi-valued+function">multi-valued function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> (or a <a class="existingWikiWord" href="/nlab/show/decidable+subset">decidable</a> <a class="existingWikiWord" href="/nlab/show/subset">subset</a> thereof) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, that is a <a class="existingWikiWord" href="/nlab/show/span">span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>←</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \leftarrow A \rightarrow X</annotation></semantics></math> where the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">A \to \mathbb{N}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a> (or has a decidable range). Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> inherits the structure of a directed set via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">A \to \mathbb{N}</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to X</annotation></semantics></math> is a net. As a net, every sequential net is equivalent (in the sense of corresponding to the same <a class="existingWikiWord" href="/nlab/show/filter">filter</a>) to some sequence, if you assume WCC. Without WCC, however, this equivalence fails.</p> <p>(Using a multi-valued function here is a special case of an alternative definition of <a class="existingWikiWord" href="/nlab/show/net">net</a> that uses only <a class="existingWikiWord" href="/nlab/show/partially+ordered">partially ordered</a> directed sets; see <a class="existingWikiWord" href="/nlab/show/net">net</a>. In some <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a>, we can get the same result by defining a sequential net to be a <strong>presequence</strong>: a <a class="existingWikiWord" href="/nlab/show/prefunction">prefunction</a>, which is like a function but need not preserve <a class="existingWikiWord" href="/nlab/show/equality">equality</a>, from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> or a decidable subset thereof.)</p> <p>Without WCC, many of the usual properties of <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> and other <a class="existingWikiWord" href="/nlab/show/sequential+spaces">sequential spaces</a> fail, but they continue to hold using sequential nets in the place of sequences. For example, every (located Dedekind) <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> may be represented as a sequential Cauchy net, even when they might not all be represented as Cauchy sequences; see <a class="existingWikiWord" href="/nlab/show/Cauchy+real+number">Cauchy real number</a>.</p> <h2 id="sequence_types">Sequence types</h2> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, a sequence type is simply the <a class="existingWikiWord" href="/nlab/show/function+type">function type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \to A</annotation></semantics></math>, and thus comes with the following rules:</p> <p>Formation rules for sequence types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathbb{N} \to A \; \mathrm{type}}</annotation></semantics></math></div> <p>Introduction rules for sequence types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>.</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N} \vdash a(n):A}{\Gamma \vdash \lambda(n:\mathbb{N}).a(n):\mathbb{N} \to A}</annotation></semantics></math></div> <p>Elimination rules for sequence types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi mathvariant="normal">ev</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, n:\mathbb{N} \vdash \mathrm{ev}(a, n):A}</annotation></semantics></math></div> <p>Computation rules for sequence types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><msub><mi>β</mi> <mi>Π</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>:</mo><mi mathvariant="normal">ev</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>.</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>a</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N} \vdash a(n):A}{\Gamma, m:\mathbb{N} \vdash \beta_\Pi(m):\mathrm{ev}(\lambda(n:\mathbb{N}).a(n), m) =_{A} a(m)}</annotation></semantics></math></div> <p>Uniqueness rules for sequence types:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi><mo>⊢</mo><msub><mi>η</mi> <mi>Π</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>:</mo><mi>a</mi><msub><mo>=</mo> <mrow><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow></msub><mi>λ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>.</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A \vdash \eta_\Pi(a):a =_{\mathbb{N} \to A} \lambda(n:\mathbb{N}).a(n)}</annotation></semantics></math></div> <p>Sequence types also have their own extensionality principle, called <a class="existingWikiWord" href="/nlab/show/sequence+extensionality">sequence extensionality</a>. This states that given two sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{N} \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{N} \to A</annotation></semantics></math> there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> between the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mo>=</mo> <mrow><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow></msub><mi>b</mi></mrow><annotation encoding="application/x-tex">a =_{\mathbb{N} \to A} b</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/dependent+sequence+type">dependent sequence type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n:\mathbb{N}) \to (a(n) =_{A} b(n))</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>A</mi><mo>⊢</mo><mi mathvariant="normal">seqext</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mo>=</mo> <mrow><mi>ℕ</mi><mo>→</mo><mi>A</mi></mrow></msub><mi>b</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>b</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, b:\mathbb{N} \to A \vdash \mathrm{seqext}(a, b):(a =_{\mathbb{N} \to A} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A} b(n))}</annotation></semantics></math></div> <p>Sequence types are used in <a class="existingWikiWord" href="/nlab/show/strongly+predicative+mathematics">strongly predicative mathematics</a>, where one does not have <a class="existingWikiWord" href="/nlab/show/function+types">function types</a>, to construct the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> <h2 id="sequence_spaces">Sequence spaces</h2> <p>In <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>, one considers <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a> of infinite sequences; these are the <a class="existingWikiWord" href="/nlab/show/sequence+spaces">sequence spaces</a>. (Actually, these generalise quite nicely to <a class="existingWikiWord" href="/nlab/show/net">net</a> spaces.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+space">sequentially compact space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence+algebra">sequence algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequential+net">sequential net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tuple">tuple</a>, <strong>sequence</strong>, <a class="existingWikiWord" href="/nlab/show/function">function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+tuple">dependent tuple</a>, <a class="existingWikiWord" href="/nlab/show/dependent+sequence">dependent sequence</a>, <a class="existingWikiWord" href="/nlab/show/dependent+function">dependent function</a></p> </li> </ul> <p>Not all that related, but similar sounding:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+limit">sequential limit</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 9, 2023 at 23:27:59. 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