Auxiliary normed space

This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (April 2020) (Learn how and when to remove this template message)

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.^[1] One method is used if the disk ${\displaystyle D}$ is bounded: in this case, the auxiliary normed space is ${\displaystyle \operatorname {span} D}$ with norm ${\displaystyle p_{D}(x):=\inf _{x\in rD,r>0}r.}$ The other method is used if the disk ${\displaystyle D}$ is absorbing: in this case, the auxiliary normed space is the quotient space ${\displaystyle X/p_{D}^{-1}(0).}$ If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces).

Preliminaries[]

A subset of a vector space is called a disk and is said to be disked, absolutely convex, or convex balanced if it is convex and balanced.

If ${\displaystyle C}$ and ${\displaystyle D}$ are subsets of a vector space ${\displaystyle X}$ then ${\displaystyle D}$ absorbs ${\displaystyle C}$ if there exists a real ${\displaystyle r>0}$ such that ${\displaystyle C\subseteq aD}$for any scalar ${\displaystyle a}$ satisfying ${\displaystyle |a|\geq r.}$ WThe set ${\displaystyle D}$ is called absorbing in ${\displaystyle X}$ if ${\displaystyle D}$ absorbs ${\displaystyle \{x\}}$ for every ${\displaystyle x\in X.}$

A subset ${\displaystyle B}$ of a topological vector space (TVS) ${\displaystyle X}$ is said to be bounded in ${\displaystyle X}$ if every neighborhood of the origin in ${\displaystyle X}$ absorbs ${\displaystyle B.}$ A subset of a TVS ${\displaystyle X}$ is called bornivorous^[2] if it absorbs all bounded subsets of ${\displaystyle X.}$

Induced by a bounded disk – Banach disks[]

Henceforth, ${\displaystyle X}$ will be a real or complex vector space (not necessarily a TVS, yet) and ${\displaystyle D}$ will be a disk in ${\displaystyle X.}$

Seminormed space induced by a disk[]

Let ${\displaystyle X}$ will be a real or complex vector space. For any subset ${\displaystyle D}$ of ${\displaystyle X,}$ the Minkowski functional of ${\displaystyle D}$ defined by:

• If ${\displaystyle D=\varnothing }$ then define ${\displaystyle p_{\varnothing }(x):\{0\}\to [0,\infty )}$ to be the trivial map ${\displaystyle p_{\varnothing }=0}$^[3] and it will be assumed that ${\displaystyle \operatorname {span} \varnothing =\{0\}.}$^{[note 1]}
• If ${\displaystyle D\neq \varnothing }$ and if ${\displaystyle D}$ is absorbing in ${\displaystyle \operatorname {span} D}$ then denote the Minkowski functional of ${\displaystyle D}$ in ${\displaystyle \operatorname {span} D}$ by

  ${\displaystyle p_{D}:\operatorname {span} D\to [0,\infty )}$

  where for all ${\displaystyle x\in \operatorname {span} D,}$ this is defined by

  ${\displaystyle p_{D}(x):=\inf _{}\left\{r:x\in rD,r>0\right\}.}$

Let ${\displaystyle X}$ will be a real or complex vector space. For any subset ${\displaystyle D}$ of ${\displaystyle X}$ such that the Minkowski functional ${\displaystyle p_{D}}$is a seminorm on ${\displaystyle \operatorname {span} D,}$ let ${\displaystyle X_{D}}$ denote

${\displaystyle X_{D}:=\left(\operatorname {span} D,p_{D}\right)}$

which is called the seminormed space induced by ${\displaystyle D}$ where it is say "normed" if ${\displaystyle p_{D}}$is a norm.

Assumption (Topology): ${\displaystyle X_{D}=\operatorname {span} D}$ is endowed with the seminorm topology induced by ${\displaystyle p_{D},}$ which will be denoted by ${\displaystyle \tau _{D}}$ or ${\displaystyle \tau _{p_{D}}}$

Importantly, this topology stems entirely from the set ${\displaystyle D,}$ the algebraic structure of ${\displaystyle X,}$ and the usual topology on ${\displaystyle \mathbb {R} }$ (since ${\displaystyle p_{D}}$is defined using only the set ${\displaystyle D}$ and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of nuclear operators and nuclear spaces.

The inclusion map ${\displaystyle \operatorname {In} _{D}:X_{D}\to X}$ is called the canonical map.^[1]

Suppose that ${\displaystyle D}$ is a disk. Then ${\displaystyle \operatorname {span} D=\bigcup _{n=1}^{\infty }nD}$ so that ${\displaystyle D}$ is absorbing in ${\displaystyle \operatorname {span} D,}$ the linear span of ${\displaystyle D.}$ The set ${\displaystyle \{rD:r>0\}}$ of all positive scalar multiples of ${\displaystyle D}$ forms a basis of neighborhoods at 0 for a locally convex topological vector space topology ${\displaystyle \tau _{D}}$ on ${\displaystyle \operatorname {span} D.}$ The Minkowski functional of the disk ${\displaystyle D}$ in ${\displaystyle \operatorname {span} D}$ guarantees that ${\displaystyle p_{D}}$is well-defined and forms a seminorm on ${\displaystyle \operatorname {span} D.}$^[4] The locally convex topology topology induced by this seminorm is the topology ${\displaystyle \tau _{D}}$ that was defined before.

Banach disk definition[]

A bounded disk ${\displaystyle D}$ in a topological vector space ${\displaystyle X}$ such that ${\displaystyle \left(X_{D},p_{D}\right)}$ is a Banach space is called a Banach disk, infracomplete, or a bounded completant in ${\displaystyle X.}$

If its shown that ${\displaystyle \left(\operatorname {span} D,p_{D}\right)}$ is a Banach space then ${\displaystyle D}$ will be a Banach disk in any TVS that contains ${\displaystyle D}$ as a bounded subset.

This is because the Minkowski functional ${\displaystyle p_{D}}$is defined in purely algebraic terms. Consequently, the question of whether or not ${\displaystyle \left(X_{D},p_{D}\right)}$ forms a Banach space is dependent only on the disk ${\displaystyle D}$ and the Minkowski functional ${\displaystyle p_{D},}$ and not on any particular TVS topology that ${\displaystyle X}$ may carry. Thus the requirement that a Banach disk in a TVS ${\displaystyle X}$ be a bounded subset of ${\displaystyle X}$ is the only property that ties a Banach disk's topology to the topology of its containing TVS ${\displaystyle X.}$

Properties of disk induced seminormed spaces[]

Bounded disks

The following result explains why Banach disks are required to be bounded.

Hausdorffness

The space ${\displaystyle \left(X_{D},p_{D}\right)}$ is Hausdorff if and only if ${\displaystyle p_{D}}$is a norm, which happens if and only if ${\displaystyle D}$ does not contain any non-trivial vector subspace.^[6] In particular, if there exists a Hausdorff TVS topology on ${\displaystyle X}$ such that ${\displaystyle D}$ is bounded in ${\displaystyle X}$ then ${\displaystyle p_{D}}$is a norm. An example where ${\displaystyle X_{D}}$ is not Hausdorff is obtained by letting ${\displaystyle X=\mathbb {R} ^{2}}$ and letting ${\displaystyle D}$ be the ${\displaystyle x}$-axis.

Convergence of nets

Suppose that ${\displaystyle D}$ is a disk in ${\displaystyle X}$ such that ${\displaystyle X_{D}}$ is Hausdorff and let ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ be a net in ${\displaystyle X_{D}.}$ Then ${\displaystyle x_{\bullet }\to 0}$ in ${\displaystyle X_{D}}$ if and only if there exists a net ${\displaystyle r_{\bullet }=\left(r_{i}\right)_{i\in I}}$ of real numbers such that ${\displaystyle r_{\bullet }\to 0}$ and ${\displaystyle x_{i}\in r_{i}D}$ for all ${\displaystyle i}$; moreover, in this case it will be assumed without loss of generality that ${\displaystyle r_{i}\geq 0}$ for all ${\displaystyle i.}$

Relationship between disk-induced spaces

If ${\displaystyle C\subseteq D\subseteq X}$then ${\displaystyle \operatorname {span} C\subseteq \operatorname {span} D}$ and ${\displaystyle p_{D}\leq p_{C}}$ on ${\displaystyle \operatorname {span} C,}$ so define the following continuous^[2] linear map:

If ${\displaystyle C}$ and ${\displaystyle D}$ are disks in ${\displaystyle X}$ with ${\displaystyle C\subseteq D}$ then call the inclusion map ${\displaystyle \operatorname {In} _{C}^{D}:X_{C}\to X_{D}}$ the canonical inclusion of ${\displaystyle X_{C}}$ into ${\displaystyle X_{D}.}$

In particular, the subspace topology that ${\displaystyle \operatorname {span} C}$ inherits from ${\displaystyle \left(X_{D},p_{D}\right)}$ is weaker than ${\displaystyle \left(X_{C},p_{C}\right)}$'s seminorm topology.^[2]

${\displaystyle D}$ as the closed unit ball

The disk ${\displaystyle D}$ is a closed subset of ${\displaystyle \left(X_{D},p_{D}\right)}$ if and only if ${\displaystyle D}$ is the closed unit ball of the seminorm ${\displaystyle p_{D}}$; that is, ${\displaystyle D=\left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}.}$

If ${\displaystyle D}$ is a disk in a vector space ${\displaystyle X}$ and if there exists a TVS topology ${\displaystyle \tau }$ on ${\displaystyle \operatorname {span} D}$ such that ${\displaystyle D}$ is a closed and bounded subset of ${\displaystyle \left(\operatorname {span} D,\tau \right),}$ then ${\displaystyle D}$ is the closed unit ball of ${\displaystyle \left(X_{D},p_{D}\right)}$ (that is, ${\displaystyle D=\left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}}$ ) (see footnote for proof).^{[note 2]}

Sufficient conditions for a Banach disk[]

The following theorem may be used to establish that ${\displaystyle \left(X_{D},p_{D}\right)}$ is a Banach space. Once this is established, ${\displaystyle D}$ will be a Banach disk in any TVS in which ${\displaystyle D}$ is bounded.

Note that even if ${\displaystyle D}$ is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that ${\displaystyle \left(X_{D},p_{D}\right)}$ is a Banach space by applying this theorem to some disk

${\displaystyle K}$ satisfying ${\displaystyle \left\{x\in \operatorname {span} D:p_{D}(x)<1\right\}\subseteq K\subseteq \left\{x\in \operatorname {span} D:p_{D}(x)\leq 1\right\}}$

because ${\displaystyle p_{D}=p_{K}.}$

The following are consequences of the above theorem:

• A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk.^[2]
• Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk.^[8]
• The closed unit ball in a Fréchet space is sequentially complete and thus a Banach disk.^[2]

Suppose that ${\displaystyle D}$ is a bounded disk in a TVS ${\displaystyle X.}$

• If ${\displaystyle L:X\to Y}$ is a continuous linear map and ${\displaystyle B\subseteq X}$ is a Banach disk, then ${\displaystyle L(B)}$ is a Banach disk and ${\displaystyle L{\big \vert }_{X_{B}}:X_{B}\to L\left(X_{B}\right)}$ induces an isometric TVS-isomorphism ${\displaystyle Y_{L(B)}\cong X_{B}/\left(X_{B}\cap \operatorname {ker} L\right).}$

Properties of Banach disks[]

Let ${\displaystyle X}$ be a TVS and let ${\displaystyle D}$ be a bounded disk in ${\displaystyle X.}$

If ${\displaystyle D}$ is a bounded Banach disk in a Hausdorff locally convex space ${\displaystyle X}$ and if ${\displaystyle T}$ is a barrel in ${\displaystyle X}$ then ${\displaystyle T}$ absorbs ${\displaystyle D}$ (i.e. there is a number ${\displaystyle r>0}$ such that ${\displaystyle D\subseteq rT.}$^[5]

If ${\displaystyle U}$ is a convex balanced closed neighborhood of 0 in ${\displaystyle X}$ then the collection of all neighborhoods ${\displaystyle rU,}$ where ${\displaystyle r>0}$ ranges over the positive real numbers, induces a topological vector space topology on ${\displaystyle X.}$ When ${\displaystyle X}$ has this topology, it is denoted by ${\displaystyle X_{U}.}$ Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space ${\displaystyle X/p_{U}^{-1}(0)}$ is denoted by ${\displaystyle {\overline {X_{U}}}}$ so that ${\displaystyle {\overline {X_{U}}}}$ is a complete Hausdorff space and ${\displaystyle p_{U}(x):=\inf _{x\in rU,r>0}r}$ is a norm on this space making ${\displaystyle {\overline {X_{U}}}}$ into a Banach space. The polar of ${\displaystyle U,}$ ${\displaystyle U^{\circ },}$ is a weakly compact bounded equicontinuous disk in ${\displaystyle X^{\prime }}$ and so is infracomplete.

If ${\displaystyle X}$ is a metrizable locally convex TVS then for every bounded subset ${\displaystyle B}$ of ${\displaystyle X,}$ there exists a bounded disk ${\displaystyle D}$ in ${\displaystyle X}$ such that ${\displaystyle B\subseteq X_{D},}$ and both ${\displaystyle X}$ and ${\displaystyle X_{D}}$ induce the same subspace topology on ${\displaystyle B.}$^[2]

Induced by a radial disk – quotient[]

Suppose that ${\displaystyle X}$ is a topological vector space and ${\displaystyle V}$ is a convex balanced and radial set. Then ${\displaystyle \left\{{\frac {1}{n}}V:n=1,2,\ldots \right\}}$ is a neighborhood basis at the origin for some locally convex topology ${\displaystyle \tau _{V}}$ on ${\displaystyle X.}$ This TVS topology ${\displaystyle \tau _{V}}$ is given by the Minkowski functional formed by ${\displaystyle V,}$ ${\displaystyle p_{V}:X\to \mathbb {R} ,}$ which is a seminorm on ${\displaystyle X}$ defined by ${\displaystyle p_{V}(x):=\inf _{x\in rV,r>0}r.}$ The topology ${\displaystyle \tau _{V}}$ is Hausdorff if and only if ${\displaystyle p_{V}}$ is a norm, or equivalently, if and only if ${\displaystyle X/p_{V}^{-1}(0)=\{0\}}$ or equivalently, for which it suffices that ${\displaystyle V}$ be bounded in ${\displaystyle X.}$ The topology ${\displaystyle \tau _{V}}$ need not be Hausdorff but ${\displaystyle X/p_{V}^{-1}(0)}$ is Hausdorff. A norm on ${\displaystyle X/p_{V}^{-1}(0)}$ is given by ${\displaystyle \left\|x+X/p_{V}^{-1}(0)\right\|:=p_{V}(x),}$ where this value is in fact independent of the representative of the equivalence class ${\displaystyle x+X/p_{V}^{-1}(0)}$ chosen. The normed space ${\displaystyle \left(X/p_{V}^{-1}(0),\|\cdot \|\right)}$ is denoted by ${\displaystyle X_{V}}$ and its completion is denoted by ${\displaystyle {\overline {X_{V}}}.}$

If in addition ${\displaystyle V}$ is bounded in ${\displaystyle X}$ then the seminorm ${\displaystyle p_{V}:X\to \mathbb {R} }$ is a norm so in particular, ${\displaystyle p_{V}^{-1}(0)=\{0\}.}$ In this case, we take ${\displaystyle X_{V}}$ to be the vector space ${\displaystyle X}$ instead of ${\displaystyle X/\{0\}}$ so that the notation ${\displaystyle X_{V}}$ is unambiguous (whether ${\displaystyle X_{V}}$ denotes the space induced by a radial disk or the space induced by a bounded disk).^[1]

The quotient topology ${\displaystyle \tau _{Q}}$ on ${\displaystyle X/p_{V}^{-1}(0)}$ (inherited from ${\displaystyle X}$'s original topology) is finer (in general, strictly finer) than the norm topology.

Canonical maps[]

The canonical map is the quotient map ${\displaystyle q_{V}:X\to X_{V}=X/p_{V}^{-1}(0),}$ which is continuous when ${\displaystyle X_{V}}$ has either the norm topology or the quotient topology.^[1]

If ${\displaystyle U}$ and ${\displaystyle V}$ are radial disks such that ${\displaystyle U\subseteq V}$then ${\displaystyle p_{U}^{-1}(0)\subseteq p_{V}^{-1}(0)}$ so there is a continuous linear surjective canonical map ${\displaystyle q_{V,U}:X/p_{U}^{-1}(0)\to X/p_{V}^{-1}(0)=X_{V}}$ defined by sending ${\displaystyle x+p_{U}^{-1}(0)\in X_{U}=X/p_{U}^{-1}(0)}$ to the equivalence class ${\displaystyle x+p_{V}^{-1}(0),}$ where one may verify that the definition does not depend on the representative of the equivalence class ${\displaystyle x+p_{U}^{-1}(0)}$ that is chosen.^[1] This canonical map has norm ${\displaystyle \,\leq 1}$^[1] and it has a unique continuous linear canonical extension to ${\displaystyle {\overline {X_{U}}}}$ that is denoted by ${\displaystyle {\overline {g_{V,U}}}:{\overline {X_{U}}}\to {\overline {X_{V}}}.}$

Suppose that in addition ${\displaystyle B\neq \varnothing }$ and ${\displaystyle C}$ are bounded disks in ${\displaystyle X}$ with ${\displaystyle B\subseteq C}$ so that ${\displaystyle X_{B}\subseteq X_{C}}$ and the inclusion ${\displaystyle \operatorname {In} _{B}^{C}:X_{B}\to X_{C}}$ is a continuous linear map. Let ${\displaystyle \operatorname {In} _{B}:X_{B}\to X,}$ ${\displaystyle \operatorname {In} _{C}:X_{C}\to X,}$ and ${\displaystyle \operatorname {In} _{B}^{C}:X_{B}\to X_{C}}$ be the canonical maps. Then ${\displaystyle \operatorname {In} _{C}=\operatorname {In} _{B}^{C}\circ \operatorname {In} _{C}:X_{B}\to X_{C}}$ and ${\displaystyle q_{V}=q_{V,U}\circ q_{U}.}$^[1]

Induced by a bounded radial disk[]

Suppose that ${\displaystyle S}$ is a bounded radial disk. Since ${\displaystyle S}$ is a bounded disk, if ${\displaystyle D:=S}$ then we may create the auxiliary normed space ${\displaystyle X_{D}=\operatorname {span} D}$ with norm ${\displaystyle p_{D}(x):=\inf _{x\in rD,r>0}r}$; since ${\displaystyle S}$ is radial, ${\displaystyle X_{S}=X.}$ Since ${\displaystyle S}$ is a radial disk, if ${\displaystyle V:=S}$ then we may create the auxiliary seminormed space ${\displaystyle X/p_{V}^{-1}(0)}$ with the seminorm ${\displaystyle p_{V}(x):=\inf _{x\in rV,r>0}r}$; because ${\displaystyle S}$ is bounded, this seminorm is a norm and ${\displaystyle p_{V}^{-1}(0)=\{0\}}$ so ${\displaystyle X/p_{V}^{-1}(0)=X/\{0\}=X.}$ Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.

Duality[]

Suppose that ${\displaystyle H}$ is a weakly closed equicontinuous disk in ${\displaystyle X^{\prime }}$ (this implies that ${\displaystyle H}$ is weakly compact) and let

${\displaystyle U:=H^{\circ }=\left\{x\in X:|h(x)|\leq 1{\text{ for all }}h\in H\right\}}$

be the polar of ${\displaystyle H.}$ Because ${\displaystyle U^{\circ }=H^{\circ \circ }=H}$ by the bipolar theorem, it follows that a continuous linear functional ${\displaystyle f}$ belongs to ${\displaystyle X_{H}^{\prime }=\operatorname {span} H}$ if and only if ${\displaystyle f}$ belongs to the continuous dual space of ${\displaystyle \left(X,p_{U}\right),}$ where ${\displaystyle p_{U}}$ is the Minkowski functional of ${\displaystyle U}$ defined by ${\displaystyle p_{U}(x):=\inf _{x\in rU,r>0}r.}$^[9]

Related concepts[]

A disk in a TVS is called infrabornivorous^[2] if it absorbs all Banach disks.

A linear map between two TVSs is called infrabounded^[2] if it maps Banach disks to bounded disks.

Fast convergence[]

A sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in a TVS ${\displaystyle X}$ is said to be fast convergent^[2] to a point ${\displaystyle x\in X}$ if there exists a Banach disk ${\displaystyle D}$ such that both ${\displaystyle x}$ and the sequence is (eventually) contained in ${\displaystyle \operatorname {span} D}$ and ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle \left(X_{D},p_{D}\right).}$

Every fast convergent sequence is Mackey convergent.^[2]

See also[]

• Bornological space – A topological vector space where any bounded linear operator into another space is always continuous
• Injective tensor product
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Nuclear operator
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Initial topology – coarsest topology making certain functions continuous
• Projective tensor product
• Schwartz topological vector space
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces
• Topological vector space – Vector space with a notion of nearness
• Ultrabornological space

Notes[]

References[]

Bibliography[]

• Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
• (1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. 720. Berlin New York: Springer-Verlag. ISBN 978-3-540-09504-0. OCLC 5126156.
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
• (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. OCLC 316549583.
• ; (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• ; (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: . ISBN 978-0-387-05644-9. OCLC 539541.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. . 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• (2002). Introduction to Tensor Products of Banach Spaces. . London New York: Springer. ISBN 978-1-85233-437-6. OCLC 48092184.
• Schaefer, Helmut H.; (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.

External links[]

• Nuclear space at ncatlab