Projective tensor product

<h2 id="definitions">Definitions</h2>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be locally convex topological vector spaces. Their projective tensor product 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 is the unique locally convex topological vector space with underlying vector space 
 
 
 
 X
 ⊗
 Y
 
 
 {\displaystyle X\otimes Y}
 
 having the following <a href="/facts/Universal_property/5XJBYMNz">universal property</a>:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

For any locally convex topological vector space 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
, if 
 
 
 
 
 Φ
 
 Z
 
 
 
 
 {\displaystyle \Phi _{Z}}
 
 is the canonical map from the vector space of bilinear maps 
 
 
 
 X
 ×
 Y
 →
 Z
 
 
 {\displaystyle X\times Y\to Z}
 
 to the vector space of linear maps 
 
 
 
 X
 ⊗
 Y
 →
 Z
 
 
 {\displaystyle X\otimes Y\to Z}
 
, then the image of the restriction of 
 
 
 
 
 Φ
 
 Z
 
 
 
 
 {\displaystyle \Phi _{Z}}
 
 to the continuous bilinear maps is the space of continuous linear maps 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 →
 Z
 
 
 {\displaystyle X\otimes _{\pi }Y\to Z}
 
.
When the topologies of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are induced by <a href="/facts/Seminorm/vMuescKp">seminorms</a>, the topology of 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 is induced by seminorms constructed from those on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 as follows. If 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is a seminorm on 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 is a seminorm on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, define their tensor product 
 
 
 
 p
 ⊗
 q
 
 
 {\displaystyle p\otimes q}
 
 to be the seminorm on 
 
 
 
 X
 ⊗
 Y
 
 
 {\displaystyle X\otimes Y}
 
 given by

(
        p
        ⊗
        q
        )
        (
        b
        )
        =
        
          inf
          
            r
            >
            0
            ,
            
            b
            ∈
            r
            W
          
        
        r
      
    
    {\displaystyle (p\otimes q)(b)=\inf _{r>0,\,b\in rW}r}

for all 
 
 
 
 b
 
 
 {\displaystyle b}
 
 in 
 
 
 
 X
 ⊗
 Y
 
 
 {\displaystyle X\otimes Y}
 
, where 
 
 
 
 W
 
 
 {\displaystyle W}
 
 is the <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> convex hull of the set 
 
 
 
 
 {
 
 x
 ⊗
 y
 :
 p
 (
 x
 )
 ≤
 1
 ,
 q
 (
 y
 )
 ≤
 1
 
 }
 
 
 
 {\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}}
 
. The projective topology on 
 
 
 
 X
 ⊗
 Y
 
 
 {\displaystyle X\otimes Y}
 
 is generated by the collection of such tensor products of the seminorms on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
When 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Normed_space/rmDljfyE">normed spaces</a>, this definition applied to the norms on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 gives a norm, called the projective norm, on 
 
 
 
 X
 ⊗
 Y
 
 
 {\displaystyle X\otimes Y}
 
 which generates the projective topology.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="properties">Properties</h2>
Throughout, all spaces are assumed to be locally convex. The symbol 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\pi }Y}
 
 denotes the completion of the projective tensor product of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.

<ul><li>If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are both <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> then so is 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
;<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> then 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 is <a href="/facts/Barrelled_space/r2H58TN3">barelled</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>For any two continuous linear operators 
 
 
 
 
 u
 
 1
 
 
 :
 
 X
 
 1
 
 
 →
 
 Y
 
 1
 
 
 
 
 {\displaystyle u_{1}:X_{1}\to Y_{1}}
 
 and 
 
 
 
 
 u
 
 2
 
 
 :
 
 X
 
 2
 
 
 →
 
 Y
 
 2
 
 
 
 
 {\displaystyle u_{2}:X_{2}\to Y_{2}}
 
, their tensor product (as linear maps) 
 
 
 
 
 u
 
 1
 
 
 ⊗
 
 u
 
 2
 
 
 :
 
 X
 
 1
 
 
 
 ⊗
 
 π
 
 
 
 X
 
 2
 
 
 →
 
 Y
 
 1
 
 
 
 ⊗
 
 π
 
 
 
 Y
 
 2
 
 
 
 
 {\displaystyle u_{1}\otimes u_{2}:X_{1}\otimes _{\pi }X_{2}\to Y_{1}\otimes _{\pi }Y_{2}}
 
 is continuous.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>In general, the projective tensor product does not respect subspaces (e.g. if 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 is a vector subspace of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 then the TVS 
 
 
 
 Z
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle Z\otimes _{\pi }Y}
 
 has in general a <a href="/facts/Comparison_of_topologies/5Q9F2fjf">coarser</a> topology than the subspace topology inherited from 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
).<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>If 
 
 
 
 E
 
 
 {\displaystyle E}
 
 and 
 
 
 
 F
 
 
 {\displaystyle F}
 
 are <a href="/facts/Complemented_subspace/v8koLMPI">complemented subspaces</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 respectively, then 
 
 
 
 E
 ⊗
 F
 
 
 {\displaystyle E\otimes F}
 
 is a complemented vector subspace of 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 and the projective norm on 
 
 
 
 E
 
 ⊗
 
 π
 
 
 F
 
 
 {\displaystyle E\otimes _{\pi }F}
 
 is equivalent to the projective norm on 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 restricted to the subspace 
 
 
 
 E
 ⊗
 F
 
 
 {\displaystyle E\otimes F}
 
. Furthermore, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 F
 
 
 {\displaystyle F}
 
 are complemented by projections of norm 1, then 
 
 
 
 E
 ⊗
 F
 
 
 {\displaystyle E\otimes F}
 
 is complemented by a projection of norm 1.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>Let 
 
 
 
 E
 
 
 {\displaystyle E}
 
 and 
 
 
 
 F
 
 
 {\displaystyle F}
 
 be vector subspaces of the <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, respectively. Then 
 
 
 
 E
 
 
 
 ⊗
 ^
 
 
 
 F
 
 
 {\displaystyle E{\widehat {\otimes }}F}
 
 is a TVS-subspace of 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\pi }Y}
 
 if and only if every bounded bilinear form on 
 
 
 
 E
 ×
 F
 
 
 {\displaystyle E\times F}
 
 extends to a continuous bilinear form on 
 
 
 
 X
 ×
 Y
 
 
 {\displaystyle X\times Y}
 
 with the same norm.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li></ul>
<h2 id="completion">Completion</h2>

In general, the space 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 is not complete, even if both 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are complete (in fact, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are both infinite-dimensional Banach spaces then 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 is necessarily not complete<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>). However, 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
 can always be linearly embedded as a <a href="/facts/Dense_set/nPT9Yxao">dense</a> vector subspace of some complete locally convex TVS, which is generally denoted by 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\pi }Y}
 
.
The <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> of 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\pi }Y}
 
 is the same as that of 
 
 
 
 X
 
 ⊗
 
 π
 
 
 Y
 
 
 {\displaystyle X\otimes _{\pi }Y}
 
, namely, the space of continuous bilinear forms 
 
 
 
 B
 (
 X
 ,
 Y
 )
 
 
 {\displaystyle B(X,Y)}
 
.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h3>Grothendieck's representation of elements in the completion</h3>
In a Hausdorff locally convex space 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 a sequence 
 
 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is absolutely convergent if 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 ∞
 
 
 p
 
 (
 
 x
 
 i
 
 
 )
 
 <
 ∞
 
 
 {\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty }
 
 for every continuous seminorm 
 
 
 
 p
 
 
 {\displaystyle p}
 
 on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> We write 
 
 
 
 x
 =
 
 ∑
 
 i
 =
 1
 
 
 ∞
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x=\sum _{i=1}^{\infty }x_{i}}
 
 if the sequence of partial sums 
 
 
 
 
 
 (
 
 
 ∑
 
 i
 =
 1
 
 
 n
 
 
 
 x
 
 i
 
 
 
 )
 
 
 n
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }}
 
 converges to 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
The following fundamental result in the theory of topological tensor products is due to <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

Theorem—Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be metrizable locally convex TVSs and let 
 
 
 
 z
 ∈
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 .
 
 
 {\displaystyle z\in X{\widehat {\otimes }}_{\pi }Y.}
 
 Then 
 
 
 
 z
 
 
 {\displaystyle z}
 
 is the sum of an <a href="/facts/Absolutely_convergent/BXz9kBJ5">absolutely convergent</a> series

z
        =
        
          ∑
          
            i
            =
            1
          
          
            ∞
          
        
        
          λ
          
            i
          
        
        
          x
          
            i
          
        
        ⊗
        
          y
          
            i
          
        
      
    
    {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}}

where 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 ∞
 
 
 
 |
 
 
 λ
 
 i
 
 
 
 |
 
 <
 ∞
 ,
 
 
 {\displaystyle \sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,}
 
 and 
 
 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}
 
 and 
 
 
 
 
 
 (
 
 y
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }}
 
 are <a href="/facts/Null_sequence/Ktge2RwL">null sequences</a> in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 respectively.

The next theorem shows that it is possible to make the representation of 
 
 
 
 z
 
 
 {\displaystyle z}
 
 independent of the sequences 
 
 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}
 
 and 
 
 
 
 
 
 (
 
 y
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 .
 
 
 {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }.}

Theorem<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>—Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet spaces</a> and let 
 
 
 
 U
 
 
 {\displaystyle U}
 
 (resp. 
 
 
 
 V
 
 
 {\displaystyle V}
 
) be a balanced open neighborhood of the origin in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (resp. in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
). Let 
 
 
 
 
 K
 
 0
 
 
 
 
 {\displaystyle K_{0}}
 
 be a compact subset of the convex balanced hull of 
 
 
 
 U
 ⊗
 V
 :=
 {
 u
 ⊗
 v
 :
 u
 ∈
 U
 ,
 v
 ∈
 V
 }
 .
 
 
 {\displaystyle U\otimes V:=\{u\otimes v:u\in U,v\in V\}.}
 
 There exists a compact subset 
 
 
 
 
 K
 
 1
 
 
 
 
 {\displaystyle K_{1}}
 
 of the unit ball in 
 
 
 
 
 ℓ
 
 1
 
 
 
 
 {\displaystyle \ell ^{1}}
 
 and sequences 
 
 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}
 
 and 
 
 
 
 
 
 (
 
 y
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }}
 
 contained in 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and 
 
 
 
 V
 ,
 
 
 {\displaystyle V,}
 
 respectively, converging to the origin such that for every 
 
 
 
 z
 ∈
 
 K
 
 0
 
 
 
 
 {\displaystyle z\in K_{0}}
 
 there exists some 
 
 
 
 
 
 (
 
 λ
 
 i
 
 
 )
 
 
 i
 =
 1
 
 
 ∞
 
 
 ∈
 
 K
 
 1
 
 
 
 
 {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1}}
 
 such that

z
        =
        
          ∑
          
            i
            =
            1
          
          
            ∞
          
        
        
          λ
          
            i
          
        
        
          x
          
            i
          
        
        ⊗
        
          y
          
            i
          
        
        .
      
    
    {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.}

<h3>Topology of bi-bounded convergence</h3>
Let 
 
 
 
 
 
 
 B
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathfrak {B}}_{X}}
 
 and 
 
 
 
 
 
 
 B
 
 
 
 Y
 
 
 
 
 {\displaystyle {\mathfrak {B}}_{Y}}
 
 denote the families of all bounded subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 ,
 
 
 {\displaystyle Y,}
 
 respectively. Since the continuous dual space of 
 
 
 
 X
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle X{\widehat {\otimes }}_{\pi }Y}
 
 is the space of continuous bilinear forms 
 
 
 
 B
 (
 X
 ,
 Y
 )
 ,
 
 
 {\displaystyle B(X,Y),}
 
 we can place on 
 
 
 
 B
 (
 X
 ,
 Y
 )
 
 
 {\displaystyle B(X,Y)}
 
 the topology of uniform convergence on sets in 
 
 
 
 
 
 
 B
 
 
 
 X
 
 
 ×
 
 
 
 B
 
 
 
 Y
 
 
 ,
 
 
 {\displaystyle {\mathfrak {B}}_{X}\times {\mathfrak {B}}_{Y},}
 
 which is also called the topology of bi-bounded convergence. This topology is coarser than the <a href="/facts/Strong_dual_space/4RI1SYl6">strong topology</a> on 
 
 
 
 B
 (
 X
 ,
 Y
 )
 
 
 {\displaystyle B(X,Y)}
 
, and in (Grothendieck 1955), <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset 
 
 
 
 B
 ⊆
 X
 
 
 
 ⊗
 ^
 
 
 
 Y
 ,
 
 
 {\displaystyle B\subseteq X{\widehat {\otimes }}Y,}
 
 do there exist bounded subsets 
 
 
 
 
 B
 
 1
 
 
 ⊆
 X
 
 
 {\displaystyle B_{1}\subseteq X}
 
 and 
 
 
 
 
 B
 
 2
 
 
 ⊆
 Y
 
 
 {\displaystyle B_{2}\subseteq Y}
 
 such that 
 
 
 
 B
 
 
 {\displaystyle B}
 
 is a subset of the closed convex hull of 
 
 
 
 
 B
 
 1
 
 
 ⊗
 
 B
 
 2
 
 
 :=
 {
 
 b
 
 1
 
 
 ⊗
 
 b
 
 2
 
 
 :
 
 b
 
 1
 
 
 ∈
 
 B
 
 1
 
 
 ,
 
 b
 
 2
 
 
 ∈
 
 B
 
 2
 
 
 }
 
 
 {\displaystyle B_{1}\otimes B_{2}:=\{b_{1}\otimes b_{2}:b_{1}\in B_{1},b_{2}\in B_{2}\}}
 
?
Grothendieck proved that these topologies are equal when 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are both Banach spaces or both are <a href="/facts/DF-space/csjJmkad">DF-spaces</a> (a class of spaces introduced by Grothendieck<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>). They are also equal when both spaces are Fréchet with one of them being nuclear.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

<h3>Strong dual and bidual</h3>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a locally convex topological vector space and let 
 
 
 
 
 X
 
 ′
 
 
 
 
 {\displaystyle X^{\prime }}
 
 be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

Theorem<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> (Grothendieck)—Let 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 be locally convex topological vector spaces with 
 
 
 
 N
 
 
 {\displaystyle N}
 
 <a href="/facts/Nuclear_space/4FLPWdfp">nuclear</a>. Assume that both 
 
 
 
 N
 
 
 {\displaystyle N}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are Fréchet spaces, or else that they are both <a href="/facts/DF-space/csjJmkad">DF-spaces</a>. Then, denoting strong dual spaces with a subscripted 
 
 
 
 b
 
 
 {\displaystyle b}
 
:

<ol><li>The strong dual of 
 
 
 
 N
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle N{\widehat {\otimes }}_{\pi }Y}
 
 can be identified with 
 
 
 
 
 N
 
 b
 
 
 ′
 
 
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 
 Y
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }}
 
;</li>
<li>The bidual of 
 
 
 
 N
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle N{\widehat {\otimes }}_{\pi }Y}
 
 can be identified with 
 
 
 
 N
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 
 Y
 
 ′
 ′
 
 
 
 
 {\displaystyle N{\widehat {\otimes }}_{\pi }Y^{\prime \prime }}
 
;</li>
<li>If 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is reflexive then 
 
 
 
 N
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 Y
 
 
 {\displaystyle N{\widehat {\otimes }}_{\pi }Y}
 
 (and hence 
 
 
 
 
 N
 
 b
 
 
 ′
 
 
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 
 Y
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime }}
 
) is a <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive space</a>;</li>
<li>Every separately continuous bilinear form on 
 
 
 
 
 N
 
 b
 
 
 ′
 
 
 ×
 
 Y
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle N_{b}^{\prime }\times Y_{b}^{\prime }}
 
 is continuous;</li>
<li>Let 
 
 
 
 L
 
 (
 
 
 X
 
 b
 
 
 ′
 
 
 ,
 Y
 
 )
 
 
 
 {\displaystyle L\left(X_{b}^{\prime },Y\right)}
 
 be the space of bounded linear maps from 
 
 
 
 
 X
 
 b
 
 
 ′
 
 
 
 
 {\displaystyle X_{b}^{\prime }}
 
 to 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. Then, its strong dual can be identified with 
 
 
 
 
 N
 
 b
 
 
 ′
 
 
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 
 Y
 
 b
 
 
 ′
 
 
 ,
 
 
 {\displaystyle N_{b}^{\prime }{\widehat {\otimes }}_{\pi }Y_{b}^{\prime },}
 
 so in particular if 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is reflexive then so is 
 
 
 
 
 L
 
 b
 
 
 
 (
 
 
 X
 
 b
 
 
 ′
 
 
 ,
 Y
 
 )
 
 .
 
 
 {\displaystyle L_{b}\left(X_{b}^{\prime },Y\right).}
 
</li></ol>

<h2 id="examples">Examples</h2>
<ul><li>For 
 
 
 
 (
 X
 ,
 
 
 A
 
 
 ,
 μ
 )
 
 
 {\displaystyle (X,{\mathcal {A}},\mu )}
 
 a measure space, let 
 
 
 
 
 L
 
 1
 
 
 
 
 {\displaystyle L^{1}}
 
 be the real <a href="/facts/Lp_space/gbad0Rv1">Lebesgue space</a> 
 
 
 
 
 L
 
 1
 
 
 (
 μ
 )
 
 
 {\displaystyle L^{1}(\mu )}
 
; let 
 
 
 
 E
 
 
 {\displaystyle E}
 
 be a real Banach space. Let 
 
 
 
 
 L
 
 E
 
 
 1
 
 
 
 
 {\displaystyle L_{E}^{1}}
 
 be the completion of the space of simple functions 
 
 
 
 X
 →
 E
 
 
 {\displaystyle X\to E}
 
, modulo the subspace of functions 
 
 
 
 X
 →
 E
 
 
 {\displaystyle X\to E}
 
 whose pointwise norms, considered as functions 
 
 
 
 X
 →
 
 R
 
 
 
 {\displaystyle X\to \mathbb {R} }
 
, have integral 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 with respect to 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
. Then 
 
 
 
 
 L
 
 E
 
 
 1
 
 
 
 
 {\displaystyle L_{E}^{1}}
 
 is isometrically isomorphic to 
 
 
 
 
 L
 
 1
 
 
 
 
 
 
 ⊗
 ^
 
 
 
 
 π
 
 
 E
 
 
 {\displaystyle L^{1}{\widehat {\otimes }}_{\pi }E}
 
.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Inductive_tensor_product/vEn3kR1T">Inductive tensor product</a> – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Injective_tensor_product/53zV1QtL">Injective tensor product</a></li>
<li><a href="/facts/Tensor_product_of_Hilbert_spaces/GSavLJfU">Tensor product of Hilbert spaces</a> – Tensor product space endowed with a special inner product</li></ul>
<h2 id="citations">Citations</h2>

<ul><li>Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-85233-437-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/48092184">48092184</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li>
<li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4440-3. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/185095773">185095773</a>.</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexander</a> (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0075539">0075539</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/9308061">9308061</a>.</li>
<li>Grothendieck, Grothendieck (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1216-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1315788">1315788</a>.</li>
<li>Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-05644-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/539541">539541</a>.</li>
<li>Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-09513-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/5126158">5126158</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/nuclear+space">Nuclear space at ncatlab</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Trèves 2006, p. 438. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Trèves 2006, p. 435. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Trèves 2006, p. 438. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Trèves 2006, p. 437. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Trèves 2006, p. 437. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Trèves 2006, p. 445. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Trèves 2006, p. 439. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Ryan 2002, p. 18. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. <a href="https://search.worldcat.org/oclc/48092184" target="_blank">https://search.worldcat.org/oclc/48092184</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Ryan 2002, p. 18. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. <a href="https://search.worldcat.org/oclc/48092184" target="_blank">https://search.worldcat.org/oclc/48092184</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Ryan 2002, p. 24. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. <a href="https://search.worldcat.org/oclc/48092184" target="_blank">https://search.worldcat.org/oclc/48092184</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Ryan 2002, p. 43. - Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. <a href="https://search.worldcat.org/oclc/48092184" target="_blank">https://search.worldcat.org/oclc/48092184</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Schaefer & Wolff 1999, p. 173. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Schaefer & Wolff 1999, p. 120. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Schaefer & Wolff 1999, p. 120. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Schaefer & Wolff 1999, p. 94. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Trèves 2006, pp. 459–460. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Schaefer & Wolff 1999, p. 154. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Schaefer & Wolff 1999, p. 173. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Schaefer & Wolff 1999, pp. 175–176. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Schaefer & Wolff 1999, p. 95. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
</ol>

Projective tensor product open-in-new

Projective tensor product