Nuclear operator

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
Preliminaries and notation.
Throughout let "X","Y", and "Z" be topological vector spaces (TVSs) and "L" : "X" → "Y" be a linear operator (no assumption of continuity is made unless otherwise stated).
A canonical tensor product as a subspace of the dual of Bi(X, Y).
Let "X" and "Y" be vector spaces (no topology is needed yet) and let Bi("X", "Y") be the space of all bilinear maps defined on formula_52 and going into the underlying scalar field.
For every formula_53, let formula_54 be the canonical bilinear form on Bi("X", "Y") defined by formula_55 for every "u" ∈ Bi("X", "Y"). 
This induces a canonical map formula_56 defined by formula_57, where formula_58 denotes the algebraic dual of Bi("X", "Y"). 
If we denote the span of the range of "𝜒" by "X" ⊗ "Y" then it can be shown that "X" ⊗ "Y" together with "𝜒" forms a tensor product of "X" and "Y" (where "x" ⊗ "y := "𝜒"("x", "y")). 
This gives us a canonical tensor product of "X" and "Y".
If "Z" is any other vector space then the mapping Li("X" ⊗ "Y"; "Z") → Bi("X", "Y"; "Z") given by "u" ↦ "u" ∘ "𝜒" is an isomorphism of vector spaces. 
In particular, this allows us to identify the algebraic dual of "X" ⊗ "Y" with the space of bilinear forms on "X" × "Y". 
Moreover, if "X" and "Y" are locally convex topological vector spaces (TVSs) and if "X" ⊗ "Y" is given the 𝜋-topology then for every locally convex TVS "Z", this map restricts to a vector space isomorphism formula_59 from the space of "continuous" linear mappings onto the space of "continuous" bilinear mappings. 
In particular, the continuous dual of "X" ⊗ "Y" can be canonically identified with the space B("X", "Y") of continuous bilinear forms on "X" × "Y"; 
furthermore, under this identification the equicontinuous subsets of B("X", "Y") are the same as the equicontinuous subsets of "formula_60.
Nuclear operators between Banach spaces.
There is a canonical vector space embedding formula_61 defined by sending formula_62 to the map
Assuming that "X" and "Y" are Banach spaces, then the map formula_64 has norm formula_65 (to see that the norm is formula_66, note that formula_67 so that formula_68). Thus it has a continuous extension to a map formula_69, where it is known that this map is not necessarily injective. The range of this map is denoted by formula_70 and its elements are called nuclear operators. formula_70 is TVS-isomorphic to formula_72 and the norm on this quotient space, when transferred to elements of formula_70 via the induced map formula_74, is called the trace-norm and is denoted by formula_75. Explicitely, if formula_76 is a nuclear operator then formula_77.
Characterization.
Suppose that "X" and "Y" are Banach spaces and that formula_78 is a continuous linear operator. 
Properties.
Let "X" and "Y" be Banach spaces and let formula_78 be a continuous linear operator. 
Nuclear operators between Hilbert spaces.
Nuclear automorphisms of a Hilbert space are called trace class operators.
Let "X" and "Y" be Hilbert spaces and let "N" : "X" → "Y" be a continuous linear map. 
Suppose that formula_99 where "R" : "X" → "X" is the square-root of formula_100 and "U" : "X" → "Y" is such that formula_30 is a surjective isometry and formula_99. 
Then "N" is a nuclear map if and only if "R" is a nuclear map; 
hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive linear operators.
Characterizations.
Let "X" and "Y" be Hilbert spaces and let "N" : "X" → "Y" be a continuous linear map whose absolute value is "R" : "X" → "X". 
The following are equivalent:
Nuclear operators between locally convex spaces.
Suppose that "U" is a convex balanced closed neighborhood of the origin in "X" and "B" is a convex balanced bounded Banach disk in "Y" with both "X" and "Y" locally convex spaces. Let formula_122 and let formula_123 be the canonical projection. One can define the auxiliary Banach space formula_124 with the canonical map formula_125 whose image, formula_126, is dense in formula_124 as well as the auxiliary space formula_128 normed by formula_129 and with a canonical map formula_130 being the (continuous) canonical injection. 
Given any continuous linear map formula_131 one obtains through composition the continuous linear map formula_132; thus we have an injection formula_133 and we henceforth use this map to identify formula_134 as a subspace of formula_135.
Definition: Let "X" and "Y" be Hausdorff locally convex spaces. The union of all formula_136 as "U" ranges over all closed convex balanced neighborhoods of the origin in "X" and "B" ranges over all bounded Banach disks in "Y", is denoted by formula_70 and its elements are call nuclear mappings of "X" into "Y".
When "X" and "Y" are Banach spaces, then this new definition of "nuclear mapping" is consistent with the original one given for the special case where "X" and "Y" are Banach spaces.
Characterizations.
Let "X" and "Y" be Hausdorff locally convex spaces and let formula_78 be a continuous linear operator. 
Properties.
The following is a type of "Hahn-Banach theorem" for extending nuclear maps:
Let "X" and "Y" be Hausdorff locally convex spaces and let formula_78 be a continuous linear operator. 