Locally convex space

A Hausdorff topological vector space over the field of real or complex numbers in which any neighbourhood of the zero element contains a convex neighbourhood of the zero element; in other words, a topological vector space $ E $ is a locally convex space if and only if the topology of $ E $ is a Hausdorff locally convex topology. Examples of locally convex spaces (and at the same time classes of locally convex spaces that are important in the theory and applications) are normed spaces, countably-normed spaces and Fréchet spaces (cf. Normed space; Countably-normed space; Fréchet space).

A number of general properties of locally convex spaces follows immediately from the corresponding properties of locally convex topologies; in particular, subspaces and Hausdorff quotient spaces of a locally convex space, and also products of families of locally convex spaces, are themselves locally convex spaces. Let $ A $ be an upward directed set of indices and $ \{ {E _ \alpha } : {\alpha \in A } \} $ a family of locally convex spaces (over the same field) with topologies $ \{ {\tau _ \alpha } : {\alpha \in A } \} $; suppose that for any pair $ ( \alpha , \beta ) $, $ \alpha \leq \beta $, $ \alpha , \beta \in A $, there is defined a continuous linear mapping $ g _ {\alpha \beta } : E _ \beta \rightarrow E _ \alpha $; let $ E $ be the subspace of the product $ \prod _ {\alpha \in A } E _ \alpha $ whose elements $ x = ( x _ \alpha ) $ satisfy the relations $ x _ \alpha = g _ {\alpha \beta } ( x _ \beta ) $ for all $ \alpha \leq \beta $; the space $ E $ is called the projective limit of the family $ \{ E _ \alpha \} $ with respect to $ \{ g _ {\alpha \beta } \} $ and is denoted by $ \lim\limits g _ {\alpha \beta } E _ \beta $ or $ \lim\limits _ \leftarrow E _ \alpha $; the topology of $ E $ is the projective topology with respect to the family $ \{ E _ \alpha , \tau _ \alpha , f _ \alpha \} $, where $ f _ \alpha $ is the restriction to the subspace $ E $ of the projection $ ( \prod _ {\beta \in A } E _ \beta ) \rightarrow E _ \alpha $. On the other hand, suppose that for any pair $ ( \alpha , \beta ) $, $ \alpha \leq \beta $, $ \alpha , \beta \in A $, there is defined a continuous linear mapping $ h _ {\alpha \beta } : E _ \alpha \rightarrow E _ \beta $; let $ g _ \alpha $, $ \alpha \in A $, be the canonical imbedding of $ E _ \alpha $ in the direct sum $ \oplus _ {\alpha \in A } E _ \alpha $ and let $ H $ be the subspace of $ \oplus _ {\alpha \in A } E _ \alpha $ generated by the images of all spaces $ E _ \alpha $ under the mappings $ g _ \alpha - g _ \beta \circ h _ {\alpha \beta } $, where $ ( \alpha , \beta ) $ runs through all pairs in $ A \times A $ for which $ \alpha \leq \beta $. If $ H $ is closed in $ \oplus _ {\alpha \in A } E _ \alpha $, then the locally convex space $ ( \oplus _ {\alpha \in A } E _ \alpha ) / H $ is called the inductive limit of the family $ \{ E _ \alpha \} $ with respect to $ \{ h _ {\alpha \beta } \} $, and is denoted by $ \lim\limits h _ {\alpha \beta } E _ \alpha $ or $ \lim\limits _ \rightarrow E _ \alpha $. If $ \{ E _ \alpha \} $ is a family of subspaces of a vector space $ E $, ordered by inclusion, and the topology $ \tau _ \beta $ induces $ \tau _ \alpha $ on $ E _ \alpha $ for $ \alpha \leq \beta $, then the inductive limit of the family $ \{ E _ \alpha \} $ is said to be strict. A locally convex space is metrizable if and only if its topology is induced by a sequence of semi-norms (cf. Semi-norm); a locally convex space is normable if and only if it contains a bounded open set (Kolmogorov's theorem). Any finite-dimensional subspace of a locally convex space has a complemented closed subspace. The completion of a locally convex space is a locally convex space, and any complete locally convex space is isomorphic to the projective limit of a family of Banach spaces. The space $ L ( F , E ) $ of continuous linear mappings from a topological vector space $ F $ into a locally convex space $ E $ is naturally endowed with the structure of a locally convex space (see also Operator topology) with respect to a given family $ \gamma $ of bounded subsets of $ F $ for which the linear hull of its union is dense in $ F $. A basis of neighbourhoods of zero of the corresponding topology is the family of sets $ \{ {f } : {f \in L ( F , E ), f ( S) \subset V } \} $, where $ S $ runs through $ \gamma $ and $ V $ runs through a basis of neighbourhoods of zero in $ E $.

A central topic in the theory of locally convex spaces (and also in the theory of topological vector spaces) is the study of the relation of the space with its dual or adjoint space. The foundation of this theory of duality for locally convex spaces is the Hahn–Banach theorem, which implies, in particular, that if $ E $ is a locally convex space, then its dual space $ E ^ \prime $ separates the points of $ E $.

An essential part of the theory of locally convex spaces is the theory of compact convex sets in a locally convex space. The convex hull $ \mathop{\rm co} K $ and the convex balanced hull (cf. also Balanced set) of a pre-compact set $ K $ in a locally convex space $ E $ are pre-compact; if $ E $ is also quasi-complete, then the closed convex hull $ { \mathop{\rm co} } bar K $ of $ K $ and its closed convex balanced hull are compact. If $ A $ and $ K $ are disjoint non-empty convex subsets of a locally convex space $ E $, where $ A $ is closed and $ K $ is compact, then there is a continuous real linear functional $ f $ on $ E $ such that for some real number $ \alpha $ the inequalities $ f ( x) > \alpha $, $ f ( y) < \alpha $ hold for all $ x \in A $, $ y \in K $, respectively. In particular, a non-empty closed convex set $ A $ in a locally convex space is the intersection of all closed half-spaces containing it. A non-empty closed convex subset $ B $ of a closed convex set $ A $ is called a face (or extremal subset) of $ A $ if any closed segment in $ A $ with an interior point in $ B $ lies entirely in $ B $; a point $ x \in A $ is called an extreme point of $ A $ if the set $ \{ x \} $ is a face of $ A $. If $ K $ is a compact convex set in a locally convex space $ E $ and $ \partial _ {e} K $ is the set of its extreme points, then the following conditions are equivalent for a set $ X \subset K $: 1) $ { \mathop{\rm co} } bar X = K $; 2) $ \overline{X}\; \supset \partial _ {e} K $; and 3) $ \sup f ( X) = \sup f ( K) $ for any continuous real linear functional $ f $ on $ E $. In particular, $ { \mathop{\rm co} } bar \partial _ {e} K = K $( the Krein–Mil'man theorem). The set $ \partial _ {e} K $ is a Baire space in the induced topology (that is, the intersection of any sequence of open subsets of $ \partial _ {e} K $ that are dense in $ \partial _ {e} K $ is dense in $ \partial K $), and for any $ x \in K $ there is a probability measure $ \mu $ on $ K $ such that $ x = \int _ {K} d \mu ( y) $ and the measure $ \mu $ vanishes on all Baire subsets $ X \subset K $ that do not intersect $ \partial _ {e} K $( if $ K $ is metrizable, then $ \mu ( \partial _ {e} K ) = 1 $) (Choquet's theorem). Any continuous mapping of a compact convex set $ K $ in a locally convex space into itself has a fixed point (the Schauder–Tikhonov theorem); a commuting family of continuous affine transformations of $ K $ onto itself (and an equicontinuous group of continuous affine transformations of $ K $ onto itself) has a fixed point (the Markov–Kakutani theorem).

One quite important branch of the theory of locally convex spaces is the theory of linear operators on a locally convex space; in particular, the theory of compact (also called completely-continuous), nuclear and Fredholm operators (cf. Compact operator; Fredholm operator; Nuclear operator). Closed-graph and open-mapping theorems have far-reaching generalizations in the theory of locally convex spaces. A locally convex space $ E $ is said to have the approximation property if the identity mapping of $ E $ into itself can be uniformly approximated on pre-compact sets of $ E $ by finite-rank continuous linear mappings of $ E $ into itself. If a locally convex space has the approximation property, then it has a number of other remarkable properties. In particular, in such a space any nuclear operator has a uniquely defined trace. There are separable Banach spaces that do not have the approximation property, but Banach spaces with a Schauder basis and subspaces of projective limits of Hilbert spaces do have the approximation property. Some versions of this property are of interest in the theory of completely-continuous and Fredholm operators.

A notable role in the theory of locally convex spaces is played by methods of homological algebra connected with the study of the category of locally convex spaces and their continuous mappings, and also some subcategories of this category. In particular, homological methods have made it possible to solve a number of problems connected with the extension of linear mappings and with the existence of a linear mapping into a given space that lifts a mapping into a quotient space of this space, and also to study properties of completions of quotient spaces $ E / F $ in relation to the completions of the spaces $ E $ and $ F $.

Other important questions in the theory of locally convex spaces are: the theory of integration of vector-valued functions with values in a locally convex space (as a rule, a barrelled space); the theory of differentiation of non-linear mappings between locally convex spaces; the theory of topological tensor products of locally convex spaces and the theory of Fredholm operators and nuclear operators. There is a detailed theory of a number of special classes of locally convex spaces, such as a barrelled spaces (cf. Barrelled space), bornological spaces (on which any semi-norm that is bounded on bounded sets is continuous), reflexive and semi-reflexive spaces (the canonical mapping of which into the strong second dual is a topological or linear isomorphism, respectively), nuclear spaces (cf. Nuclear space), etc.

References

[1]	A.N. Kolmogorov, "Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes" Studia Math. , 5 (1935) pp. 29–33
[2]	N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[3]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[4]	H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[5]	M.M. Day, "Normed linear spaces" , Springer (1973)
[6]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[7]	A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[8]	W.J. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1973)
[9]	R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966)
[10]	P. Enflo, "A counterexample to the approximation problem in Banach spaces" Acta. Math. , 130 (1973) pp. 309–317
[11]	A. Frölicher, W. Bucher, "Calculus in vector spaces without norm" , Lect. notes in math. , 30 , Springer (1966)
[12]	V.P. Palmodov, "Homological methods in the theory of locally convex spaces" Russian Math. Surveys , 26 : 1 (1971) pp. 1–64 Uspekhi Mat. Nauk , 26 : 1 (1971) pp. 3–65

Comments

Locally convex spaces arise in great profusion throughout such fields of analysis as measure and integration theory, complex analysis in one, several or an infinite number of variables, partial differential equations, integral equations, approximation theory, operator and spectral theory, as well as probability theory. Many sequence spaces, spaces of holomorphic, continuous or measurable functions, spaces of measures, test functions and distributions have a natural locally convex topology.

The powerful duality theory of locally convex spaces provides an important tool to translate a problem on the space (or on linear operators between locally convex spaces) into one concerning the linear forms. The fundamental results of duality theory include the bipolar theorem (a form of the Hahn–Banach theorem), the Alaoğlu–Bourbaki theorem (on equicontinuous sets in the dual) and the Mackey–Arens theorem (characterizing the topologies which are compatible with a given dual pair). By means of duality theory, the surjectivity of linear operators and the existence of continuous linear right inverses (leading to solution operators for partial differential equations) can be studied; V.P. Palamodov developed the homological methods with such applications in mind. There is an abstract duality between topology and bornology, and equicontinuous sets provide an important example of compactology.

Part of the classical structure theory of locally convex spaces can be viewed as a generalization of the (basic) theory of Banach spaces (cf. Banach space) and of its main theorems (which are often consequences of the Hahn–Banach theorem and of the Baire category theorem, cf. Baire theorem). This development led to the introduction of special classes of locally convex spaces, of which the most important ones are: Fréchet and (DF)-spaces, barrelled and bornological spaces, reflexive spaces, (LF)-spaces (i.e., countable inductive limits of Fréchet spaces), nuclear, Schwartz and Montel spaces.

Topological tensor products were introduced as a tool to study spaces of operators and spaces of vector-valued functions and distributions. A. Grothendieck [a4] investigated the nuclear spaces in this context and raised the approximation problem which was solved by P. Enflo [10] by giving the first example of a Banach space without the approximation property. Later on, A. Szankowski proved that the space of all bounded linear operators on a Hilbert space does not have the approximation property.

Besides compact convex sets (Choquet's theory has important applications in abstract potential theory), also weakly compact sets are studied (cf. [a3]).

References [a5]–[a8] are general monographs on locally convex spaces and duality theory. [a1], [a9] and [a10] are devoted to more specialized topics, while [a2] is a monograph on infinite-dimensional holomorphy and its connections to the theory of locally convex spaces.

References

[a1]	M. DeWilde, "Closed graph theorems and webbed spaces" , Pitman (1978)
[a2]	S. Dineen, "Complex analysis in locally convex spaces" , North-Holland (1981)
[a3]	K. Floret, "Weakly compact sets" , Lect. notes in math. , 801 , Springer
[a4]	A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[a5]	A. Grothendieck, "Topological vector spaces" , Gordon & Breach (1973) (Translated from French)
[a6]	J. Horváth, "Topological vector spaces and distributions" , I , Addison-Wesley (1966)
[a7]	H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
[a8]	G. Köthe, "Topological vector spaces" , 1–2 , Springer (1969–1979)
[a9]	P. Pérez Carreras, "Barrelled locally convex spaces" , North-Holland (1987)
[a10]	J. Schmets, "Espaces de fonctions continues" , Lect. notes in math. , 519 , Springer (1976)
[a11]	G. Choquet, "Lectures on analysis" , 1–3 , Benjamin (1969) (Translated from French)