Infunctional analysis and related areas ofmathematics,Fréchet spaces, named afterMaurice Fréchet, are specialtopological vector spaces. They are generalizations ofBanach spaces (normed vector spaces that arecomplete with respect to themetric induced by thenorm). AllBanach andHilbert spaces are Fréchet spaces. Spaces ofinfinitely differentiablefunctions are typical examples of Fréchet spaces, many of which are typicallynot Banach spaces.

A Fréchet space${\displaystyle X}$ is defined to be alocally convexmetrizabletopological vector space (TVS) that iscomplete as a TVS,^[1] meaning that everyCauchy sequence in${\displaystyle X}$ converges to some point in${\displaystyle X}$ (see footnote for more details).^{[note 1]}

Important note: Not all authors require that a Fréchet space be locally convex (discussed below).

The topology of every Fréchet space is induced by sometranslation-invariantcomplete metric. Conversely, if the topology of a locally convex space${\displaystyle X}$ is induced by a translation-invariant complete metric then${\displaystyle X}$ is a Fréchet space.

Fréchet was the first to use the term "Banach space" andBanach in turn then coined the term "Fréchet space" to mean acompletemetrizable topological vector space, without the local convexity requirement (such a space is today often called an "F-space").^[1] The local convexity requirement was added later byNicolas Bourbaki.^[1] A sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "F-space" and "Fréchet space" interchangeably. When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "F-space" and "Fréchet space" requires local convexity.^[1]

Fréchet spaces can be defined in two equivalent ways: the first employs atranslation-invariantmetric, the second acountable family ofseminorms.

A topological vector space${\displaystyle X}$ is aFréchet space if and only if it satisfies the following three properties:

1. It islocally convex.^{[note 2]}
2. Its topologycan beinduced by a translation-invariant metric, that is, a metric${\displaystyle d:X\times X\to \mathbb{R}}$ such that${\displaystyle d(x,y)=d(x+z,y+z)}$ for all${\displaystyle x,y,z\in X.}$ This means that a subset${\displaystyle U}$ of${\displaystyle X}$ isopen if and only if for every${\displaystyle u\in U}$ there exists an${\displaystyle r>0}$ such that is a subset of${\displaystyle U.}$
3. Some (or equivalently, every) translation-invariant metric on${\displaystyle X}$ inducing the topology of${\displaystyle X}$ iscomplete.
   • Assuming that the other two conditions are satisfied, this condition is equivalent to${\displaystyle X}$ being acomplete topological vector space, meaning that${\displaystyle X}$ is a completeuniform space when it is endowed with itscanonical uniformity (this canonical uniformity is independent of any metric on${\displaystyle X}$ and is defined entirely in terms of vector subtraction and${\displaystyle X}$'s neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on${\displaystyle X}$ is identical to this canonical uniformity).

Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.

The alternative and somewhat more practical definition is the following: a topological vector space${\displaystyle X}$ is aFréchet space if and only if it satisfies the following three properties:

1. It is aHausdorff space.
2. Its topology may be induced by a countable family of seminorms${\displaystyle (\Vert \cdot {\Vert }_k{)}_{k\in {\mathbb{N}}_0}}$. This means that a subset${\displaystyle U\subseteq X}$ is open if and only if for every${\displaystyle u\in U}$ there exist${\displaystyle K\ge 0}$ and${\displaystyle r>0}$ such that is a subset of${\displaystyle U}$.
3. It is complete with respect to the family of seminorms.

A family${\displaystyle \mathcal{P}}$ of seminorms on${\displaystyle X}$ yields a Hausdorff topology if and only if^[2]$${\displaystyle \bigcap _{\Vert \cdot \Vert \in \mathcal{P}}\{x\in X:\Vert x\Vert =0\}=\{0\}.}$$

A sequence${\displaystyle {\left(x_n\right)}_{n\in \mathbb{N}}}$ in${\displaystyle X}$ converges to${\displaystyle x}$ in the Fréchet space defined by a family of seminorms if and only if it converges to${\displaystyle x}$ with respect to each of the given seminorms.

In contrast toBanach spaces, the complete translation-invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both atotal paranorm and anF-norm (theF stands for Fréchet).

Even though thetopological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like theopen mapping theorem, theclosed graph theorem, and theBanach–Steinhaus theorem, still hold.

Recall that a seminorm${\displaystyle \Vert \cdot \Vert }$ is a function from a vector space${\displaystyle X}$ to the real numbers satisfying three properties. For all${\displaystyle x,y\in X}$ and all scalars${\displaystyle c,}$$${\displaystyle \Vert x\Vert \ge 0}$$$${\displaystyle \Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert }$$$${\displaystyle \Vert c\cdot x\Vert =|c|\Vert x\Vert }$$

If${\displaystyle \Vert x\Vert =0⟺x=0}$, then${\displaystyle \Vert \cdot \Vert }$ is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:

To construct a Fréchet space, one typically starts with a vector space${\displaystyle X}$ and defines a countable family of seminorms${\displaystyle \Vert \cdot {\Vert }_k}$ on${\displaystyle X}$ with the following two properties:

• if${\displaystyle x\in X}$ and${\displaystyle \Vert x{\Vert }_k=0}$ for all${\displaystyle k\ge 0,}$ then${\displaystyle x=0}$;
• if${\displaystyle x_{\bullet }={\left(x_n\right)}_{n=1}^{\mathrm{\infty }}}$ is a sequence in${\displaystyle X}$ which isCauchy with respect to each seminorm${\displaystyle \Vert \cdot {\Vert }_k,}$ then there exists${\displaystyle x\in X}$ such that${\displaystyle x_{\bullet }={\left(x_n\right)}_{n=1}^{\mathrm{\infty }}}$ converges to${\displaystyle x}$ with respect to each seminorm${\displaystyle \Vert \cdot {\Vert }_k.}$

Then the topology induced by these seminorms (as explained above) turns${\displaystyle X}$ into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete. A translation-invariant complete metric inducing the same topology on${\displaystyle X}$ can then be defined by$${\displaystyle d(x,y)=\sum _{k=0}^{\mathrm{\infty }}{2}^{-k}\frac{\Vert x-y{\Vert }_k}{1+\Vert x-y{\Vert }_k}x,y\in X.}$$

The function${\displaystyle u\mapsto \frac{u}{1+u}}$ maps${\displaystyle [0,\mathrm{\infty })}$ monotonically to${\displaystyle [0,1),}$ and so the above definition ensures that${\displaystyle d(x,y)}$ is "small" if and only if there exists${\displaystyle K}$ "large" such that${\displaystyle \Vert x-y{\Vert }_k}$ is "small" for${\displaystyle k=0,\dots ,K.}$

• Every Banach space is a Fréchet space, as the norm induces a translation-invariant metric and the space is complete with respect to this metric.
• Thespace${\displaystyle {\mathbb{R}}^{\omega }}$ of all real valued sequences (also denoted${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$) becomes a Fréchet space if we define the${\displaystyle k}$-th seminorm of a sequence to be theabsolute value of the${\displaystyle k}$-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.

• Thevector space${\displaystyle C^{\mathrm{\infty }}([0,1])}$ of all infinitely differentiable functions${\displaystyle f:[0,1]\to \mathbb{R}}$ becomes a Fréchet space with the seminorms$${\displaystyle \Vert f{\Vert }_k=sup\{|f^{(k)}(x)|:x\in [0,1]\}}$$for every non-negative integer${\displaystyle k.}$ Here,${\displaystyle f^{(k)}}$ denotes the${\displaystyle k}$-th derivative of${\displaystyle f,}$ and${\displaystyle f^{(0)}=f.}$ In this Fréchet space, a sequence${\displaystyle \left(f_n\right)\to f}$ of functionsconverges towards the element${\displaystyle f\in C^{\mathrm{\infty }}([0,1])}$ if and only if for every non-negative integer${\displaystyle k\ge 0,}$ the sequence${\displaystyle \left(f_n^{(k)}\right)\to f^{(k)}}$converges uniformly.
• The vector space${\displaystyle C^{\mathrm{\infty }}(\mathbb{R})}$ of all infinitely differentiable functions${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ becomes a Fréchet space with the seminorms$${\displaystyle \Vert f{\Vert }_{k,n}=sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$$for all integers${\displaystyle k,n\ge 0.}$ Then, a sequence of functions${\displaystyle \left(f_n\right)\to f}$ converges if and only if for every${\displaystyle k,n\ge 0,}$ the sequences${\displaystyle \left(f_n^{(k)}\right)\to f^{(k)}}$converge compactly.
• The vector space${\displaystyle C^m(\mathbb{R})}$ of all${\displaystyle m}$-times continuously differentiable functions${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ becomes a Fréchet space with the seminorms$${\displaystyle \Vert f{\Vert }_{k,n}=sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$$for all integers${\displaystyle n\ge 0}$ and${\displaystyle k=0,\dots ,m.}$
• If${\displaystyle M}$ is acompact${\displaystyle C^{\mathrm{\infty }}}$-manifold and${\displaystyle B}$ is aBanach space, then the set${\displaystyle C^{\mathrm{\infty }}(M,B)}$ of all infinitely-often differentiable functions${\displaystyle f:M\to B}$ can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If${\displaystyle M}$ is a (not necessarily compact)${\displaystyle C^{\mathrm{\infty }}}$-manifold which admits a countable sequence${\displaystyle K^n}$ of compact subsets, so that every compact subset of${\displaystyle M}$ is contained in at least one${\displaystyle K^n,}$ then the spaces${\displaystyle C^m(M,B)}$ and${\displaystyle C^{\mathrm{\infty }}(M,B)}$ are also Fréchet space in a natural manner.As a special case, every smooth finite-dimensionalcomplete manifold${\displaystyle M}$ can be made into such a nested union of compact subsets: equip it with aRiemannian metric${\displaystyle g}$ which induces a metric${\displaystyle d(x,y),}$ choose${\displaystyle x\in M,}$ and let$${\displaystyle K_n=\{y\in M:d(x,y)\le n\}.}$$Let${\displaystyle X}$ be a compact${\displaystyle C^{\mathrm{\infty }}}$-manifold and${\displaystyle V}$ avector bundle over${\displaystyle X.}$ Let${\displaystyle C^{\mathrm{\infty }}(X,V)}$ denote the space of smooth sections of${\displaystyle V}$ over${\displaystyle X.}$ Choose Riemannian metrics${\displaystyle g}$ and connections${\displaystyle D}$, which are guaranteed to exist, on the bundles${\displaystyle TX}$ and${\displaystyle V.}$ If${\displaystyle s}$ is a section, denote itsj^th covariant derivative by${\displaystyle D^{j}s.}$ Then$${\displaystyle \Vert s{\Vert }_n=\sum _{j=0}^n\underset{x\in M}{sup}|D^{j}s{|}_g}$$(where${\displaystyle |\cdot {|}_g}$ is the norm induced by the Riemannian metric${\displaystyle g}$) is a family of seminorms making${\displaystyle C^{\mathrm{\infty }}(M,V)}$ into a Fréchet space.

• Let${\displaystyle H}$ be the space of entire (everywhereholomorphic) functions on the complex plane. Then the family of seminorms$${\displaystyle {\left|f\right|}_n=sup\{\left|f(z)\right|:|z|\le n\}}$$makes${\displaystyle H}$ into a Fréchet space.
• Let${\displaystyle H}$ be the space of entire (everywhere holomorphic) functions ofexponential type${\displaystyle \tau .}$ Then the family of seminorms$${\displaystyle {\left|f\right|}_n=\underset{z\in \mathbb{C}}{sup}\exp \left[-\left(\tau +\frac{1}{n}\right)|z|\right]\left|f(z)\right|}$$makes${\displaystyle H}$ into a Fréchet space.

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is thespace${\displaystyle L^p([0,1])}$ with Although this space fails to be locally convex, it is anF-space.

If a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them. A Banach space,${\displaystyle C^{\mathrm{\infty }}([a,b]),}$${\displaystyle C^{\mathrm{\infty }}(X,V)}$ with${\displaystyle X}$ compact, and${\displaystyle H}$ all admit norms, while${\displaystyle {\mathbb{R}}^{\omega }}$ and${\displaystyle C(\mathbb{R})}$ do not.

A closed subspace of a Fréchet space is a Fréchet space. A quotient of a Fréchet space by a closed subspace is a Fréchet space. The direct sum of a finite number of Fréchet spaces is a Fréchet space.

A product ofcountably many Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if allexcept for at most countably many of them are trivial (that is, have dimension 0). Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis). So for example, if${\displaystyle I\ne \varnothing }$ is any set and${\displaystyle X}$ is any non-trivial Fréchet space (such as${\displaystyle X=\mathbb{R}}$ for instance), then the product${\displaystyle X^I=\prod _{i\in I}X}$ is a Fréchet space if and only if${\displaystyle I}$ is a countable set.

Several important tools of functional analysis which are based on theBaire category theorem remain true in Fréchet spaces; examples are theclosed graph theorem and theopen mapping theorem. Theopen mapping theorem implies that if${\displaystyle \tau \text{ and }{\tau }_2}$ are topologies on${\displaystyle X}$ that make both${\displaystyle (X,\tau )}$ and${\displaystyle \left(X,{\tau }_2\right)}$ intocompletemetrizable TVSs (such as Fréchet spaces) and if one topology isfiner or coarser than the other then they must be equal (that is, if${\displaystyle \tau \subseteq {\tau }_2\text{ or }{\tau }_2\subseteq \tau \text{ then }\tau ={\tau }_2}$).^[4]

Everybounded linear operator from a Fréchet space into anothertopological vector space (TVS) is continuous.^[5]

There exists a Fréchet space${\displaystyle X}$ having abounded subset${\displaystyle B}$ and also a dense vector subspace${\displaystyle M}$ such that${\displaystyle B}$ isnot contained in the closure (in${\displaystyle X}$) of any bounded subset of${\displaystyle M.}$^[6]

All Fréchet spaces arestereotype spaces. In the theory of stereotype spaces Fréchet spaces are dual objects toBrauner spaces. AllmetrizableMontel spaces areseparable.^[7] Aseparable Fréchet space is a Montel space if and only if eachweak-* convergent sequence in its continuous dual converges isstrongly convergent.^[7]

Thestrong dual space${\displaystyle X_b^{\mathrm{\prime }}}$ of a Fréchet space (and more generally, of any metrizable locally convex space^[8])${\displaystyle X}$ is aDF-space.^[9] The strong dual of a DF-space is a Fréchet space.^[10] The strong dual of areflexive Fréchet space is abornological space^[8] and aPtak space. Every Fréchet space is a Ptak space. The strong bidual (that is, thestrong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.^[11]

If${\displaystyle X}$ is a locally convex space then the topology of${\displaystyle X}$ can be a defined by a family of continuousnorms on${\displaystyle X}$ (anorm is apositive-definiteseminorm) if and only if there existsat least one continuousnorm on${\displaystyle X.}$^[12] Even if a Fréchet space has a topology that is defined by a (countable) family ofnorms (all norms are also seminorms), then it may nevertheless still fail to benormable space (meaning that its topology can not be defined by any single norm). Thespace of all sequences${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$ (with the product topology) is a Fréchet space. There does not exist any Hausdorfflocally convex topology on${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$ that isstrictly coarser than this product topology.^[13] The space${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$ is notnormable, which means that its topology can not be defined by anynorm.^[13] Also, there does not existanycontinuous norm on${\displaystyle {\mathbb{K}}^{\mathbb{N}}.}$ In fact, as the following theorem shows, whenever${\displaystyle X}$ is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of${\displaystyle {\mathbb{K}}^{\mathbb{N}}}$ as a subspace.

If${\displaystyle X}$ is a non-normable Fréchet space on which there exists a continuous norm, then${\displaystyle X}$ contains a closed vector subspace that has notopological complement.^[14]

A metrizablelocally convex space isnormable if and only if itsstrong dual space is aFréchet–Urysohn locally convex space.^[9] In particular, if a locally convex metrizable space${\displaystyle X}$ (such as a Fréchet space) isnot normable (which can only happen if${\displaystyle X}$ is infinite dimensional) then itsstrong dual space${\displaystyle X_b^{\mathrm{\prime }}}$ is not aFréchet–Urysohn space and consequently, thiscomplete Hausdorff locally convex space${\displaystyle X_b^{\mathrm{\prime }}}$ is also neither metrizable nor normable.

Thestrong dual space of a Fréchet space (and more generally, ofbornological spaces such as metrizable TVSs) is always acomplete TVS and so like any complete TVS, it isnormable if and only if its topology can be induced by acomplete norm (that is, if and only if it can be made into aBanach space that has the same topology). If${\displaystyle X}$ is a Fréchet space then${\displaystyle X}$ isnormable if (and only if) there exists a completenorm on its continuous dual space${\displaystyle X^{\prime }}$ such that the norm induced topology on${\displaystyle X^{\prime }}$ isfiner than the weak-* topology.^[15] Consequently, if a Fréchet space isnot normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.

Note that the homeomorphism described in the Anderson–Kadec theorem isnot necessarily linear.

If${\displaystyle X}$ and${\displaystyle Y}$ are Fréchet spaces, then the space${\displaystyle L(X,Y)}$ consisting of allcontinuouslinear maps from${\displaystyle X}$ to${\displaystyle Y}$ isnot a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, theGateaux derivative:

Suppose${\displaystyle U}$ is an open subset of a Fréchet space${\displaystyle X,}$${\displaystyle P:U\to Y}$ is a function valued in a Fréchet space${\displaystyle Y,}$${\displaystyle x\in U}$ and${\displaystyle h\in X.}$ The map${\displaystyle P}$ isdifferentiable at${\displaystyle x}$ in the direction${\displaystyle h}$ if thelimit$${\displaystyle D(P)(x)(h)=\underset{t\to 0}{lim}\frac{1}{t}\left(P(x+th)-P(x)\right)}$$exists. The map${\displaystyle P}$ is said to becontinuously differentiable in${\displaystyle U}$ if the map$${\displaystyle D(P):U\times X\to Y}$$is continuous. Since theproduct of Fréchet spaces is again a Fréchet space, we can then try to differentiate${\displaystyle D(P)}$ and define the higher derivatives of${\displaystyle P}$ in this fashion.

The derivative operator${\displaystyle P:C^{\mathrm{\infty }}([0,1])\to C^{\mathrm{\infty }}([0,1])}$ defined by${\displaystyle P(f)=f^{\prime }}$ is itself infinitely differentiable. The first derivative is given by$${\displaystyle D(P)(f)(h)=h^{\prime }}$$for any two elements${\displaystyle f,h\in C^{\mathrm{\infty }}([0,1]).}$ This is a major advantage of the Fréchet space${\displaystyle C^{\mathrm{\infty }}([0,1])}$ over the Banach space${\displaystyle C^k([0,1])}$ for finite${\displaystyle k.}$

If${\displaystyle P:U\to Y}$ is a continuously differentiable function, then thedifferential equation$${\displaystyle x^{\prime }(t)=P(x(t)),x(0)=x_0\in U}$$need not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces.

In general, theinverse function theorem is not true in Fréchet spaces, although a partial substitute is theNash–Moser theorem.

One may defineFréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look likeEuclidean space${\displaystyle {\mathbb{R}}^n}$), and one can then extend the concept ofLie group to these manifolds. This is useful because for a given (ordinary) compact${\displaystyle C^{\mathrm{\infty }}}$ manifold${\displaystyle M,}$ the set of all${\displaystyle C^{\mathrm{\infty }}}$diffeomorphisms${\displaystyle f:M\to M}$ forms a generalized Lie group in this sense, and this Lie group captures the symmetries of${\displaystyle M.}$ Some of the relations betweenLie algebras and Lie groups remain valid in this setting.

Another important example of a Fréchet Lie group is the loop group of a compact Lie group${\displaystyle G,}$ the smooth (${\displaystyle C^{\mathrm{\infty }}}$) mappings${\displaystyle \gamma :S^1\to G,}$ multiplied pointwise by${\displaystyle \left({\gamma }_1{\gamma }_2\right)(t)={\gamma }_1(t){\gamma }_2(t).}$^[16][17]

If we drop the requirement for the space to be locally convex, we obtainF-spaces: vector spaces with complete translation-invariant metrics.

LF-spaces are countable inductive limits of Fréchet spaces.

• Brauner space
• Fréchet lattice – Topological vector lattice
• Graded Fréchet space – Generalization of the inverse function theoremPages displaying short descriptions of redirect targets
• Surjection of Fréchet spaces – Characterization of surjectivity
• Tame Fréchet space – Generalization of the inverse function theoremPages displaying short descriptions of redirect targets

• "Fréchet space",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Berberian, Sterling K. (1974).Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer.ISBN 978-0-387-90081-0.OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981].Topological Vector Spaces: Chapters 1–5.Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.ISBN 3-540-13627-4.OCLC 17499190.
• Conway, John (1990).A course in functional analysis.Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York:Springer-Verlag.ISBN 978-0-387-97245-9.OCLC 21195908.
• Edwards, Robert E. (1995).Functional Analysis: Theory and Applications. New York: Dover Publications.ISBN 978-0-486-68143-6.OCLC 30593138.
• 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.
• Jarchow, Hans (1981).Locally convex spaces. Stuttgart: B.G. Teubner.ISBN 978-3-519-02224-4.OCLC 8210342.
• 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.
• Köthe, Gottfried (1983) [1969].Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.ISBN 978-3-642-64988-2.MR 0248498.OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Pressley, Andrew;Segal, Graeme (1986).Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. New York:Oxford University Press.ISBN 0-19-853535-X.MR 0900587.
• Robertson, Alex P.; Robertson, Wendy J. (1980).Topological Vector Spaces.Cambridge Tracts in Mathematics. Vol. 53. Cambridge England:Cambridge University Press.ISBN 978-0-521-29882-7.OCLC 589250.
• Rudin, Walter (1991).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.
• 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.
• Sergeev, Armen (2010).Kähler Geometry of Loop Spaces. Mathematical Society of Japan Memoirs. Vol. 23.World Scientific Publishing.doi:10.1142/e023.ISBN 978-4-931469-60-0.
• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York:Springer-Verlag.ISBN 978-3-540-08662-8.OCLC 297140003.
• Swartz, Charles (1992).An introduction to Functional Analysis. New York: M. Dekker.ISBN 978-0-8247-8643-4.OCLC 24909067.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
• Wilansky, Albert (2013).Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc.ISBN 978-0-486-49353-4.OCLC 849801114.

• Fréchet spaces
• Topological vector spaces
• F-spaces

• Articles with short description
• Short description is different from Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link