Surjection of Fréchet spaces
The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. 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. Preliminaries, definitions, and notation. Let formula_1 be a continuous linear map between topological vector spaces. The continuous dual space of formula_2 is denoted by formula_3 The transpose of formula_4 is the map formula_5 defined by formula_6 If formula_1 is surjective then formula_5 will be injective, but the converse is not true in general. The weak topology on formula_2 (resp. formula_10) is denoted by formula_11 (resp. formula_12). The set formula_2 endowed with this topology is denoted by formula_14 The topology formula_11 is the weakest topology on formula_2 making all linear functionals in formula_10 continuous. If formula_18  then the polar of formula_19 in formula_20 is denoted by formula_21 If formula_22 is a seminorm on formula_2, then formula_24 will denoted the vector space formula_2 endowed with the weakest TVS topology making formula_26 continuous. A neighborhood basis of formula_24 at the origin consists of the sets formula_28 as formula_29 ranges over the positive reals. If formula_26 is not a norm then formula_24 is not Hausdorff and formula_32 is a linear subspace of formula_2. If formula_26 is continuous then the identity map formula_35 is continuous so we may identify the continuous dual space formula_36 of formula_24 as a subset of  formula_10 via the transpose of the identity map formula_39 which is injective. Lemmas. The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces. They are useful even on their own. Applications. Linear partial differential operators. formula_40 being means that for every relatively compact open subset formula_41 of formula_42, the following condition holds: formula_42 being means that for every compact subset formula_48 and every integer formula_49 there is a compact subset formula_50 of formula_42 such that for every distribution formula_52 with compact support in formula_42, the following condition holds: