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Surjection of Fréchet spaces
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<p>The theorem on the <a href="/facts/Surjection/KezhLpCb">surjection</a> of <a href="/facts/Fr%25C3%25A9chet_spaces/9gXkQu2u">Fréchet spaces</a> is an important theorem, due to <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a>, that characterizes when a <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operator</a> between Fréchet spaces is surjective. </p><p>The importance of this theorem is related to the <a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">open mapping theorem</a>, which states that a continuous linear surjection between Fréchet spaces is an <a href="/facts/Open_map/abHUKX7l">open map</a>. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal. </p>