Schauder basis

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Schauder basis or countable basis is similar to the usual (<a href="/facts/Hamel_basis/89IPoN6c">Hamel</a>) <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>; the difference is that Hamel bases use <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> including <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>.
</p><p>Schauder bases were described by <a href="/facts/Juliusz_Schauder/VSpCQfY1">Juliusz Schauder</a> in 1927, although such bases were discussed earlier. For example, the <a href="/facts/Haar_basis/F2XIo6yq">Haar basis</a> was given in 1909, and <a href="/facts/Georg_Faber/5j4BJm3R">Georg Faber</a> discussed in 1910 a basis for <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> on an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a>, sometimes called a Faber–Schauder system.
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Schauder basis open-in-new

Schauder basis