Differentiable function in functional analysis
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of
-times continuously differentiable functions on an open subset
of Euclidean space
(
), which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space
so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.
All vector spaces will be assumed to be over the field
where
is either the real numbers
or the complex numbers
Continuously differentiable vector-valued functions
A map
which may also be denoted by
between two topological spaces is said to be
-times continuously differentiable or
if it is continuous. A topological embedding may also be called a
-embedding.
Curves
Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the Gateaux derivative. They are fundamental to the analysis of maps between two arbitrary topological vector spaces
and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.
A continuous map
from a subset
that is valued in a topological vector space
is said to be (once or
-time) differentiable if for all
it is differentiable at
which by definition means the following limit in
exists:
![{\displaystyle f^{\prime }(t):=f^{(1)}(t):=\lim _{\stackrel {r\to t}{t\neq r\in I}}{\frac {f(r)-f(t)}{r-t}}=\lim _{\stackrel {h\to 0}{t\neq t+h\in I}}{\frac {f(t+h)-f(t)}{h}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a8eade9008a9845c9c556c50170b044311fe5c)
where in order for this limit to even be well-defined,
![{\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
must be an accumulation point of
![{\displaystyle I.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d88084f0ce6b21a819684057ef0e480b900c0bc)
If
![{\displaystyle f:I\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22dac5d63b9fe8bd772a1b377e9673c59eb2d81c)
is differentiable then it is said to be
continuously differentiable or
![{\displaystyle C^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91)
if its
derivative, which is the induced map
![{\displaystyle f^{\prime }=f^{(1)}:I\to X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/123de150c0d27157dede24e4f41bf3a4504066ad)
is continuous. Using induction on
![{\displaystyle 1<k\in \mathbb {N} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2618d98d5d92b921ca06461d9089e6e106308c01)
the map
![{\displaystyle f:I\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22dac5d63b9fe8bd772a1b377e9673c59eb2d81c)
is
-times continuously differentiable or
![{\displaystyle C^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534)
if its
![{\displaystyle k-1^{\text{th}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c262893e1c1fcd54dd39764e2fa2b0e1e99cc4b6)
derivative
![{\displaystyle f^{(k-1)}:I\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d574a279c7ad082e81e0b5f8f20b9aeb74dd2977)
is continuously differentiable, in which case the
-derivative of ![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is the map
![{\displaystyle f^{(k)}:=\left(f^{(k-1)}\right)^{\prime }:I\to X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab6371b747e34f4425bb42179014cba2ab969bd)
It is called
smooth,
![{\displaystyle C^{\infty },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16c7cf8322706166398a2c611d52b988ff0824b0)
or
infinitely differentiable if it is
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
-times continuously differentiable for every integer
![{\displaystyle k\in \mathbb {N} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7037847116113f919f4a73998f7466f9c0615a1b)
For
![{\displaystyle k\in \mathbb {N} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f2d6e44ab029b7cee559d32df7149e9e221baf)
it is called
-times differentiable if it is
![{\displaystyle k-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745)
-times continuous differentiable and
![{\displaystyle f^{(k-1)}:I\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d574a279c7ad082e81e0b5f8f20b9aeb74dd2977)
is differentiable.
A continuous function
from a non-empty and non-degenerate interval
into a topological space
is called a curve or a
curve in
A path in
is a curve in
whose domain is compact while an arc or C0-arc in
is a path in
that is also a topological embedding. For any
a curve
valued in a topological vector space
is called a
-embedding if it is a topological embedding and a
curve such that
for every
where it is called a
-arc if it is also a path (or equivalently, also a
-arc) in addition to being a
-embedding.
Differentiability on Euclidean space
The definition given above for curves are now extended from functions valued defined on subsets of
to functions defined on open subsets of finite-dimensional Euclidean spaces.
Throughout, let
be an open subset of
where
is an integer. Suppose
and
is a function such that
with
an accumulation point of
Then
is differentiable at
if there exist
vectors
in
called the partial derivatives of
at
, such that
![{\displaystyle \lim _{\stackrel {p\to t}{t\neq p\in \operatorname {domain} f}}{\frac {f(p)-f(t)-\sum _{i=1}^{n}\left(p_{i}-t_{i}\right)e_{i}}{\|p-t\|_{2}}}=0{\text{ in }}Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab935f4ec0138ca52758d41e78829d52c650b2da)
where
![{\displaystyle p=\left(p_{1},\ldots ,p_{n}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd462a2bb331be5ab0000f34f369fcb790715b3c)
If
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is differentiable at a point then it is continuous at that point. If
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is differentiable at every point in some subset
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
of its domain then
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is said to be (
once or
-time)
differentiable in ![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
, where if the subset
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is not mentioned then this means that it is differentiable at every point in its domain. If
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is differentiable and if each of its partial derivatives is a continuous function then
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is said to be (
once or
-time)
continuously differentiable or
![{\displaystyle C^{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f387d15762e2a4d3582bac399af8bc09fd990089)
For
![{\displaystyle k\in \mathbb {N} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f2d6e44ab029b7cee559d32df7149e9e221baf)
having defined what it means for a function
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
to be
![{\displaystyle C^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534)
(or
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
times continuously differentiable), say that
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is
times continuously differentiable or that
is ![{\displaystyle C^{k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bf34a5752811e9d397b02efbde8f1d6cd8abd3)
if
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is continuously differentiable and each of its partial derivatives is
![{\displaystyle C^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba867d51403599603b521561e1befc27db949552)
Say that
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is
smooth,
![{\displaystyle C^{\infty },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16c7cf8322706166398a2c611d52b988ff0824b0)
or
infinitely differentiable if
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is
![{\displaystyle C^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534)
for all
![{\displaystyle k=0,1,\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed33855f38b4d3fa637e8b4054b6e5986927c3d)
The
support of a function
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is the closure (taken in its domain
![{\displaystyle \operatorname {domain} f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c263fa98e612300ed2581acc294d75e94bf2a07)
) of the set
Spaces of Ck vector-valued functions
In this section, the space of smooth test functions and its canonical LF-topology are generalized to functions valued in general complete Hausdorff locally convex topological vector spaces (TVSs). After this task is completed, it is revealed that the topological vector space
that was constructed could (up to TVS-isomorphism) have instead been defined simply as the completed injective tensor product
of the usual space of smooth test functions
with
Throughout, let
be a Hausdorff topological vector space (TVS), let
and let
be either:
- an open subset of
where
is an integer, or else - a locally compact topological space, in which case
can only be ![{\displaystyle 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962)
Space of Ck functions
For any
let
denote the vector space of all
-valued maps defined on
and let
denote the vector subspace of
consisting of all maps in
that have compact support. Let
denote
and
denote
Give
the topology of uniform convergence of the functions together with their derivatives of order
on the compact subsets of
Suppose
is a sequence of relatively compact open subsets of
whose union is
and that satisfy
for all
Suppose that
is a basis of neighborhoods of the origin in
Then for any integer
the sets:
![{\displaystyle {\mathcal {U}}_{i,\ell ,\alpha }:=\left\{f\in C^{k}(\Omega ;Y):\left(\partial /\partial p\right)^{q}f(p)\in U_{\alpha }{\text{ for all }}p\in \Omega _{i}{\text{ and all }}q\in \mathbb {N} ^{n},|q|\leq \ell \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d38d3e70573fb80fa6aeb5bbc122cfe53743242a)
form a basis of neighborhoods of the origin for
![{\displaystyle C^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2f4583ff9bfcf8edf0e2c2cd74805f76080734)
as
![{\displaystyle \ell ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4add99aac49ce70f3e2e4bf49f0a763f358e087e)
and
![{\displaystyle \alpha \in A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d584ba1104d5f4e64c91027696802e38979ea02)
vary in all possible ways. If
![{\displaystyle \Omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
is a countable union of compact subsets and
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is a
Fréchet space, then so is
![{\displaystyle C^{(}\Omega ;Y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/954c77ac59e7c43523de3242bb62c85e0f20d0db)
Note that
![{\displaystyle {\mathcal {U}}_{i,l,\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a543b19b640eee70ec1e46370fe322b9525bbe1)
is convex whenever
![{\displaystyle U_{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c507812c8cdaf4cea8d2e7e1705b495a3010a352)
is convex. If
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is
metrizable (resp.
complete,
locally convex,
Hausdorff) then so is
![{\displaystyle C^{k}(\Omega ;Y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/810822cb5ddaf40dcaae1faed5d394bb928466c1)
If
![{\displaystyle (p_{\alpha })_{\alpha \in A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9862c0c6edbab5968b7ebb264db751843e82ea58)
is a basis of continuous seminorms for
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
then a basis of continuous seminorms on
![{\displaystyle C^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2f4583ff9bfcf8edf0e2c2cd74805f76080734)
is:
![{\displaystyle \mu _{i,l,\alpha }(f):=\sup _{y\in \Omega _{i}}\left(\sum _{|q|\leq l}p_{\alpha }\left(\left(\partial /\partial p\right)^{q}f(p)\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbfb990d3b72a288c72eecfb1fd3c9e555a2dfc)
as
![{\displaystyle \ell ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4add99aac49ce70f3e2e4bf49f0a763f358e087e)
and
![{\displaystyle \alpha \in A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d584ba1104d5f4e64c91027696802e38979ea02)
vary in all possible ways.
Space of Ck functions with support in a compact subset
The definition of the topology of the space of test functions is now duplicated and generalized. For any compact subset
denote the set of all
in
whose support lies in
(in particular, if
then the domain of
is
rather than
) and give it the subspace topology induced by
If
is a compact space and
is a Banach space, then
becomes a Banach space normed by
Let
denote
For any two compact subsets
the inclusion
![{\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K;Y)\to C^{k}(L;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5572b49a8ae8f42835239f5f7ab44e7441ece0e)
is an embedding of TVSs and that the union of all
![{\displaystyle C^{k}(K;Y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffefea5d8f747d4e0ea4d4dbb268534a2772942d)
as
![{\displaystyle K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
varies over the compact subsets of
![{\displaystyle \Omega ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26)
is
Space of compactly support Ck functions
For any compact subset
let
![{\displaystyle \operatorname {In} _{K}:C^{k}(K;Y)\to C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78067ab54ce64fa71aeae22d6d6ba0bb5fa141f4)
denote the inclusion map and endow
![{\displaystyle C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201095b534fa9911c1d59da4b7b6a1bf4f6f9c1c)
with the strongest topology making all
![{\displaystyle \operatorname {In} _{K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd336f5c109408bb29d5bf9d06416304110f469)
continuous, which is known as the
final topology induced by these map. The spaces
![{\displaystyle C^{k}(K;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8483694fba4c701bf8d6f9a56673b764a10c085b)
and maps
![{\displaystyle \operatorname {In} _{K_{1}}^{K_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1995d875a771251a82bd32d1f1c9baf03b0422)
form a
direct system (directed by the compact subsets of
![{\displaystyle \Omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
) whose limit in the category of TVSs is
![{\displaystyle C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201095b534fa9911c1d59da4b7b6a1bf4f6f9c1c)
together with the injections
![{\displaystyle \operatorname {In} _{K}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5f84a7a33a3381382c14524c91aaec7f7cebbc)
The spaces
![{\displaystyle C^{k}\left({\overline {\Omega _{i}}};Y\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83c8d61840018aeb1f61e9cfcd397cf7fb296893)
and maps
![{\displaystyle \operatorname {In} _{\overline {\Omega _{i}}}^{\overline {\Omega _{j}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e0ef9ee2fbaa3f62b8b3c1bb09e119535d6c87)
also form a
direct system (directed by the total order
![{\displaystyle \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed)
) whose limit in the category of TVSs is
![{\displaystyle C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201095b534fa9911c1d59da4b7b6a1bf4f6f9c1c)
together with the injections
![{\displaystyle \operatorname {In} _{\overline {\Omega _{i}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe79a270e7020762c963d633b864715c22c320e)
Each embedding
![{\displaystyle \operatorname {In} _{K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd336f5c109408bb29d5bf9d06416304110f469)
is an embedding of TVSs. A subset
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
of
![{\displaystyle C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201095b534fa9911c1d59da4b7b6a1bf4f6f9c1c)
is a neighborhood of the origin in
![{\displaystyle C_{c}^{k}(\Omega ;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201095b534fa9911c1d59da4b7b6a1bf4f6f9c1c)
if and only if
![{\displaystyle S\cap C^{k}(K;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/179a01190f61aea07746b0ee204c085e9001935f)
is a neighborhood of the origin in
![{\displaystyle C^{k}(K;Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8483694fba4c701bf8d6f9a56673b764a10c085b)
for every compact
![{\displaystyle K\subseteq \Omega .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d45abccd7a0de82a67c86bc7ee4d51e0d94b0cd)
This direct limit topology (i.e. the final topology) on
![{\displaystyle C_{c}^{\infty }(\Omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa0ab489393b3d3a2962fcaa861528871f223f4)
is known as the
canonical LF topology.
If
is a Hausdorff locally convex space,
is a TVS, and
is a linear map, then
is continuous if and only if for all compact
the restriction of
to
is continuous. The statement remains true if "all compact
" is replaced with "all
".
Properties
Identification as a tensor product
Suppose henceforth that
is Hausdorff. Given a function
and a vector
let
denote the map
defined by
This defines a bilinear map
into the space of functions whose image is contained in a finite-dimensional vector subspace of
this bilinear map turns this subspace into a tensor product of
and
which we will denote by
Furthermore, if
denotes the vector subspace of
consisting of all functions with compact support, then
is a tensor product of
and
If
is locally compact then
is dense in
while if
is an open subset of
then
is dense in
See also
Notes
Citations
References
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- Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
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- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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- 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.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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