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| In [[mathematics]], there are usually many different ways to construct a '''topological tensor product''' of two [[topological vector space]]s. For [[Hilbert space]]s or [[nuclear space]]s there is a simple [[well-behaved]] theory of [[tensor product]]s (see [[Tensor product of Hilbert spaces]]), but for general [[Banach space]]s or [[locally convex topological vector space]] the theory is notoriously subtle.
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| ==Tensor products of Hilbert spaces==
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| :{{Main|Tensor product of Hilbert spaces}}
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| The algebraic tensor product of two Hilbert spaces ''A'' and ''B''
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| has a natural positive definite [[sesquilinear form]] (scalar product) induced by the sesquilinear forms of ''A'' and ''B''. So in particular it has a natural [[positive definite quadratic form]], and the corresponding completion is a Hilbert space ''A''⊗''B'', called the (Hilbert space) tensor product of ''A'' and ''B''.
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| If the vectors ''a<sub>i</sub>'' and ''b<sub>j</sub>'' run through [[Orthonormal basis|orthonormal bases]] of ''A'' and ''B'', then the vectors ''a<sub>i</sub>''⊗''b<sub>j</sub>'' form an orthonormal basis of ''A''⊗''B''.
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| ==Cross norms and tensor products of Banach spaces==
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| We shall use the notation from {{harv|Ryan|2002}} in this section. The obvious way to define the tensor product of two Banach spaces ''A'' and ''B'' is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.
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| If ''A'' and ''B'' are Banach spaces the algebraic tensor product of ''A'' and ''B'' means the [[tensor product]] of
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| ''A'' and ''B'' as vector spaces and is denoted by <math>A \otimes B</math>.
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| The algebraic tensor product <math>A \otimes B</math> consists of all finite sums
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| :<math>x = \sum_{i=1}^n a_i \otimes b_i</math> | |
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| where <math>n</math> is a natural number depending on <math>x</math> and
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| <math>a_i \in A</math> and <math>b_i \in B</math> for
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| <math>i = 1, \ldots, n</math>.
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| When ''A'' and ''B'' are Banach spaces, a '''cross norm''' ''p'' on the algebraic tensor product <math>A \otimes B</math>
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| is a norm satisfying the conditions
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| :<math>p(a \otimes b) = \|a\| \|b\|</math> | |
| :<math>p'(a' \otimes b') = \|a'\| \|b'\|.</math>
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| Here ''a''′ and ''b''′ are in the [[Dual space#Continuous dual space|topological dual spaces]] of ''A'' and ''B'', respectively,
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| and ''p''′ is the [[dual norm]] of ''p''. The term '''reasonable crossnorm''' is also used for the definition above.
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| There is a largest cross norm <math>\pi</math> called the projective cross norm, given by
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| :<math>\pi(x) = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i \right\}</math> | |
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| where <math>x \in A \otimes B</math>.
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| There is a smallest cross norm <math>\varepsilon</math> called the injective cross norm, given by
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| :<math>\varepsilon(x) = \sup \{ |(a'\otimes b')(x)| : a' \in A', b' \in B', \|a'\| = \|b'\| = 1 \}</math>
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| where <math>x \in A \otimes B</math>. Here ''A''′ and ''B''′ mean the topological duals of ''A'' and ''B'', respectively.
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| The completions of the algebraic tensor product in these two norms are called
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| the projective and injective tensor products, and are denoted by <math>A \hat{\otimes}_\pi B</math>
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| and <math>A \hat{\otimes}_\varepsilon B</math>.
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| The norm used for the Hilbert space tensor product is not equal to either of these norms in general.
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| Some authors denote it by σ, so the Hilbert space tensor product in the section above would be
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| <math>A \hat{\otimes}_\sigma B</math>.
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| A '''uniform crossnorm''' α is an assignment to each pair <math>(X, Y)</math> of Banach spaces of a reasonable crossnorm on <math>X \otimes Y</math> so that if <math>X</math>, <math>W</math>, <math>Y</math>, <math>Z</math> are arbitrary Banach spaces then for all (continuous linear) operators <math>S: X \to W</math> and <math>T: Y \to Z</math> the operator <math>S \otimes T : X \otimes_\alpha Y \to W \otimes_\alpha Z</math> is continuous and
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| <math>\|S \otimes T\| \leq \|S\| \|T\|</math>.
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| If ''A'' and ''B'' are two Banach spaces and α is a uniform cross norm then α defines a reasonable cross norm on the algebraic tensor product <math>A \otimes B</math>. The normed linear space obtained by equipping <math>A \otimes B</math> with that norm is denoted by <math>A \otimes_\alpha B</math>. The completion of <math>A \otimes_\alpha B</math>, which is a Banach space, is denoted by <math>A \hat{\otimes}_\alpha B</math>. The value of the norm given by α on <math>A \otimes B</math> and on the completed tensor product <math>A \hat{\otimes}_\alpha B</math> for an element ''x'' in <math>A \hat{\otimes}_\alpha B</math>
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| (or <math>A \otimes_\alpha B</math>) is denoted by <math>\alpha_{A,B}(x)</math> or <math>\alpha(x)</math>.
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| A uniform crossnorm <math>\alpha</math> is said to be '''finitely generated''' if, for every pair <math>(X, Y)</math> of Banach spaces and every <math>u \in X \otimes Y</math>,
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| :<math>\alpha(u; X \otimes Y) = \inf \{ \alpha(u ; M \otimes N) : \dim M, \dim N < \infty \}.</math>
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| A uniform crossnorm <math>\alpha</math> is '''cofinitely generated''' if, for every pair <math>(X, Y)</math> of Banach spaces and every <math>u \in X \otimes Y</math>,
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| :<math>\alpha(u) = \sup \{ \alpha((Q_E \otimes Q_F)u; (X/E) \otimes (Y/F)) : \dim X/E, \dim Y/F < \infty \}.</math>
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| A '''tensor norm''' is defined to be a finitely generated uniform crossnorm.
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| The projective cross norm <math>\pi</math> and the injective cross norm <math>\varepsilon</math> defined above are tensor norms
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| and they are called the projective tensor norm and the injective tensor norm, respectively.
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| If ''A'' and ''B'' are arbitrary Banach spaces and ''α'' is an arbitrary uniform cross norm then
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| :<math>\varepsilon_{A,B}(x) \leq \alpha_{A,B}(x) \leq \pi_{A,B}(x).</math>
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| ==Tensor products of locally convex topological vector spaces==
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| The topologies of locally convex topological vector spaces ''A'' and ''B'' are given by families of [[seminorm]]s. For each choice of seminorm
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| on ''A'' and on ''B'' we can define the corresponding family of cross norms on the algebraic tensor product ''A''⊗''B'', and by choosing one cross norm from each family we get some cross norms on ''A''⊗''B'', defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on ''A''⊗''B'' are called the projective and injective tensor products, and denoted by ''A''⊗<sub>γ</sub>''B'' and ''A''⊗<sub>λ</sub>''B''.
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| There is a natural map from ''A''⊗<sub>γ</sub>''B''
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| to ''A''⊗<sub>λ</sub>''B''.
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| If ''A'' or ''B'' is a [[nuclear space]] then the natural map from ''A''⊗<sub>γ</sub>''B''
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| to ''A''⊗<sub>λ</sub>''B'' is an isomorphism. Roughly speaking, this means that if ''A'' or ''B'' is nuclear, then there is only one sensible tensor product of ''A'' and ''B''.
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| This property characterizes nuclear spaces.
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| ==See also==
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| *[[Hilbert space]], [[Banach space]], [[Fréchet space]], [[locally convex topological vector space]], [[Nuclear space]]
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| *[[Tensor product of Hilbert spaces]]
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| *[[Fredholm kernel]]
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| *[[Projective topology]]
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| ==References==
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| *{{citation|last=Ryan|first=R.A.|title=Introduction to Tensor Products of Banach Spaces|publisher=Springer|publication-place=New York|year=2002}}.
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| *{{citation|first=A.|last=Grothendieck|authorlink=Alexander Grothendieck|title=Produits tensoriels topologiques et espaces nucléaires|journal=Memoirs of the American Mathematical Society|volume=16|year=1955}}.
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| {{Functional Analysis}}
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| [[Category:Operator theory]]
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| [[Category:Topological vector spaces]]
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| [[Category:Hilbert space]]
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The writer's name is Christy. She is truly fond of caving but she doesn't have the time lately. I am currently a journey agent. For years he's been residing in Alaska and he doesn't strategy on changing it.
My blog post :: best psychic readings (top article)