Uniform boundedness principle

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>. 
Together with the <a href="/facts/Hahn%25E2%2580%2593Banach_theorem/3AUHusvn">Hahn–Banach theorem</a> and the <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">open mapping theorem</a>, it is considered one of the cornerstones of the field. 
In its basic form, it asserts that for a family of <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operators</a> (and thus <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a>) whose domain is a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>, pointwise <a href="/facts/Bounded_set/yCldjtoy">boundedness</a> is equivalent to uniform boundedness in <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a>.
</p><p>The theorem was first published in 1927 by <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a> and <a href="/facts/Hugo_Steinhaus/jAvzyLhV">Hugo Steinhaus</a>, but it was also proven independently by <a href="/facts/Hans_Hahn_(mathematician)/fg9B0gCR">Hans Hahn</a>.
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Uniform boundedness principle open-in-new

Uniform boundedness principle