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Sequence space
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<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a sequence space is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> whose elements are infinite <a href="/facts/Sequence/gV2S4uqp">sequences</a> of <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>. Equivalently, it is a <a href="/facts/Function_space/lLUE2R1t">function space</a> whose elements are functions from the <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a> to the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>K</i> of real or complex numbers. The set of all such functions is naturally identified with the set of all possible <a href="/facts/Infinite_sequence/gV2S4uqp">infinite sequences</a> with elements in <i>K</i>, and can be turned into a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> under the operations of <a href="/facts/Pointwise_addition/dt4Mrs6w">pointwise addition</a> of functions and pointwise scalar multiplication. All sequence spaces are <a href="/facts/Linear_subspace/2wtKTgHL">linear subspaces</a> of this space. Sequence spaces are typically equipped with a <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>, or at least the structure of a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>. </p><p>The most important sequence spaces in analysis are the <i>ℓ</i><i>p</i> spaces, consisting of the <i>p</i>-power summable sequences, with the <i>p</i>-norm. These are special cases of <a href="/facts/Lp_space/gbad0Rv1">L<i>p</i> spaces</a> for the <a href="/facts/Counting_measure/TA4u39yF">counting measure</a> on the set of natural numbers. Other important classes of sequences like <a href="/facts/Convergent_sequence/Ktge2RwL">convergent sequences</a> or null sequences form sequence spaces, respectively denoted <i>c</i> and <i>c</i>0, with the <a href="/facts/Supremum_norm/5yIkjNE3">sup norm</a>. Any sequence space can also be equipped with the <a href="/facts/Topology/2EqW47cU">topology</a> of <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise convergence</a>, under which it becomes a special kind of <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> called <a href="/facts/FK-space/dyHIqqjz">FK-space</a>. </p>