Basis

of a set $ X $

A minimal subset $ B $ that generates it. Generation here means that by application of operations of a certain class $ \Omega $ to elements $ b \in B $ it is possible to obtain any element $ x \in X $. This concept is related to the concept of dependence: By means of operations from $ \Omega $ the elements of $ X $ become dependent on the elements of $ B $. Minimality means that no proper subset $ B _ {1} \subset B $ generates $ X $. In a certain sense this property causes the elements of $ B $ to be independent: None of the elements $ b \in B $ is generated by the other elements of $ B $. For instance, the set of all natural numbers $ \mathbf Z _ {0} $ has the unique element 0 as basis and is generated from it by the operation of immediate succession and its iteration. The set of all natural numbers $ >1 $ is generated by the operation of multiplication from the basis consisting of all prime numbers. A basis of the algebra of quaternions consists of the four elements $ \\{ 1, i, j, k \\} $ if the generating operations consist of addition and of multiplication by real numbers; if, in addition to these operations, one also includes multiplication of quaternions, the basis will consist of three elements only — $ \\{ 1, i, j \\} $( because $ k=ij $).

A basis of the natural numbers of order $ k $ is a subsequence $ \Omega $ of the set $ \mathbf Z _ {0} $ of natural numbers including 0, which, as a result of $ k $- fold addition to itself (the generating operation) yields all of $ \mathbf Z _ {0} $. This means that any natural number $ n $ can be represented in the form

$$ n = a _ {1} + \dots + a _ {k} , $$

where $ a _ {i} \in \Omega $. For example, every natural number is a sum of four squares of natural numbers (Lagrange's theorem), i.e. the sequence of squares is a basis of $ \mathbf Z _ {0} $ of order 4. In general, the sequence of $ m $- th powers of natural numbers is a basis of $ \mathbf Z _ {0} $( Hilbert's theorem), the order of which has been estimated by the Vinogradov method. The concept of a basis of $ \mathbf Z _ {0} $ has been generalized to the case of arbitrary sequences of numbers, i.e. functions on $ \mathbf Z _ {0} $.

A set $ X $ always contains a generating set (in the trivial case: $ X $ generates $ X $), but minimality may prove to be principally impossible (such a situation is typical of classes $ \Omega $ containing infinite-place operations, in particular in topological structures, lattices, etc.). For this reason the minimality condition is replaced by a weaker requirement: A basis is a generating set of minimal cardinality. In this context a basis $ B $ is defined as a parametrized set (or population), i.e. as a function $ b(t) $ on a set of indices $ T $ with values in $ X $, such that $ b(T) = B $; the cardinality of $ T $ is sometimes called as the dimension (or rank) of the basis of $ X $. For example, a countable everywhere-dense set $ B $ in a separable topological space $ P $ may be considered as a basis for it; $ P $ is generated from $ B $ by the closure operation (which, incidentally, is related to generation in more general cases as well, see below).

A basis for a topology of a topological space $ X $( a base) is a basis $ \mathfrak B $ of the set of all open subsets in $ X $; the generation is effected by taking unions of elements of $ \mathfrak B $.

A basis of a Boolean algebra $ \mathfrak A $( a dual base of $ \mathfrak A $ in the sense of Tarski) is a dense set $ S $( of minimal cardinality) in $ \mathfrak A $; the generation of $ \mathfrak A $ from $ S $( and hence $ S $ itself) is determined by the condition $ s \rightarrow a = \lor $( which is equivalent to $ s \subset a $), where $ s \in S $, $ a \in \mathfrak A $, $ \lor $ is the unit of $ \mathfrak A $ and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter $ \nabla $ as a set $ S $ such that for an arbitrary $ a \in \nabla $ there exists an $ s \in S $ with $ s \subset a $.

More special cases of bases of a set $ X $ are introduced according to the following procedure. Let $ B(X) $ be the Boolean algebra of $ X $, i.e. the set of all its subsets. A generating operator (or a closure operator) $ J $ is a mapping of $ B (X) $ into itself such that if $ A \subset B $, then $ J(A) \subset J(B) $; $ A \subset J(A) $; $ JJ(A) = J(A) $.

An element $ x \in X $ is generated by a set $ A $ if $ x \in J(A) $; in particular, $ A $ generates $ X $ if $ J(A) = X $. A minimal set $ B $ possessing this property is said to be a basis of $ X $ defined by the operator $ J $. A generating operator $ J $ is of finite type if, for arbitrary $ A \subset X $ and $ x \subset X $, it follows from $ x \in J(A) $ that $ x \in J(A _ {0} ) $ for a certain finite subset $ A _ {0} \subset A $; a generating operator $ J $ has the property of substitution if, for any $ y, z \in X $ and $ A \subset X $, both $ y \notin J(A) $ and $ y \in J(A \cup \\{ z \\} ) $ imply that $ z \in J(A \cup \\{ y \\} ) $. A generating operator $ J $ of finite type with the substitution property defines a dependence relation on $ X $, i.e. a subdivision of $ B(X) $ into two classes — dependent and independent sets; a set $ A $ is said to be dependent if $ y \in J(A \setminus y) $ for some $ y \in A $, and is said to be independent if $ y \notin J (A \setminus y) $ for any $ y \in A $. Therefore, $ A $ is dependent (independent) if and only if some (arbitrary) non-empty finite subset(s) $ A _ {0} \subset A $ is dependent (are independent).

For a set $ B $ to be a basis of the set $ X $ it is necessary and sufficient for $ B $ to be an independent generating set for $ X $, or else, a maximal independent set in $ X $.

If $ A $ is an arbitrary independent set, and $ C $ is an $ X $- generating set containing $ A $, then there exists a basis $ B $ in $ X $ such that $ A \subset B \subset C $. In particular, $ X $ always has a basis, and any two bases of it have the same cardinality.

In algebraic systems $ X $ an important role is played by the concept of the so-called free basis $ B $, which is characterized by the following property: Any mapping of $ B \subset X $ into any algebraic system $ Y $( of the same signature) may be extended to a (unique) (homo)morphism from $ X $ into $ Y $ or, which is the same thing, for any (homo)morphism $ \theta : X \rightarrow Y $ and any set $ A \subset X $, the generating operators $ J _ {X} $ and $ J _ {Y} $ satisfy the condition:

$$ \theta \\{ J _ {X} (A) \\} = \ J _ {Y} ( \theta \\{ A \\} ) . $$

An algebraic system with a free basis is said to be free.

A typical example is a basis of a (unitary) module $ M $ over a ring $ K $, that is, a free family of elements from $ M $ generating $ M $[3]. Here, a family $ A = \\{ {a _ {t} } : {t \in T } \\} $ of elements of a $ K $- module $ M $ is said to be free if $ \sum \xi _ {t} a _ {t} = 0 $( where $ \xi _ {t} = 0 $ for all except a finite number of indices $ t $) implies that $ \xi _ {t} = 0 $ for all $ t $, and the generation is realized by representing the elements $ x $ as linear combinations of the elements $ a _ {t} $: There exists a set (dependent on $ x $) of elements $ \xi _ {t} \in K $ such that $ \xi _ {t} = 0 $ for all except a finite number of indices $ t $, and such that the decomposition

$$ x = \sum \xi _ {t} a _ {t} $$

is valid (i.e. $ X $ is the linear envelope of $ A $). In this sense, the basis $ M $ is free basis; the converse proposition is also true. Thus, the set of periods of a doubly-periodic function $ f $ of one complex variable, which is a discrete Abelian group (and hence a module over the ring $ \mathbf Z $), has a free basis, called the period basis of $ f $; it consists of two so-called primitive periods. A period basis of an Abelian function of several complex variables is defined in a similar manner.

If $ K $ is a skew-field, all bases (in the previous sense) are free. On the contrary, there exist modules without a free basis; these include, for example, the non-principal ideals in an integral domain $ K $, considered as a $ K $- module.

A basis of a vector space $ X $ over a field $ K $ is a (free) basis of the unitary module which underlies $ X $. In a similar manner, a basis of an algebra $ A $ over a field $ K $ is a basis of the vector space underlying $ A $. All bases of a given vector space $ X $ have the same cardinality, which is equal to the cardinality of $ T $; the latter is called the algebraic dimension of $ X $. Each element $ x \in X $ can be represented as a linear combination of basis elements in a unique way. The elements $ \xi _ {t} (x) \in K $, which are linear functionals on $ X $, are called the components (coordinates) of $ x $ in the given basis $ \\{ a _ {t} \\} $.

A set $ A $ is a basis in $ X $ if and only if $ A $ is a maximal (with respect to inclusion) free set in $ X $.

The mapping

$$ \Xi : x \rightarrow \xi _ {x} (t), $$

where $ \xi _ {x} (t) = \xi _ {t} (x) $ if $ \xi _ {t} $ is the value of the $ t $- th component of $ x $ in the basis $ A $, and 0 otherwise, is called the basis mapping; it is a linear injective mapping of $ X $ into the space $ K ^ {T} $ of functions on $ T $ with values in $ K $. In this case the image $ \Xi (X) $ consists of all functions with a finite number of non-zero values (functions of finite support). This interpretation permits one to define a generalized basis of a vector space $ X $ over a field $ K $ as a bijective linear mapping from it to some subspace $ K (T) $ of the space $ K ^ {T} $ of functions on $ T $ with values in $ K $, where $ T $ is some suitably chosen set. However, unless additional restrictions (e.g. an order) and additional structures (e.g. a topology) are imposed on $ T $, and corresponding compatible conditions on $ K(T) $ are introduced, the concept of a generalized basis is seldom of use in practice.

A basis of a vector space $ X $ is sometimes called an algebraic basis; in this way it is stressed that there is no connection with additional structures on $ X $, even if they are compatible with its vector structure.

A Hamel basis is a basis of the field of real numbers $ \mathbf R $, considered as a vector space over the field of rational numbers. It was introduced by G. Hamel [4] to obtain a discontinuous solution of the functional equation $ f(x+y) = f(x) + f(y) $; the graph of its solution is everywhere dense in the plane $ \mathbf R ^ {2} $. To each almost-periodic function corresponds some countable Hamel basis $ \beta $ such that each Fourier exponent $ \Lambda _ {n} $ of this function belongs to the linear envelope of $ \beta $. The elements of $ \beta $ may be so chosen that they belong to a sequence $ \\{ \Lambda _ {i} \\} $; the set $ \beta $ is said to be a basis of the almost-periodic functions. An analogous basis has been constructed in a ring containing a skew-field $ P $ and which has the unit of $ P $ as its own unit. An algebraic basis of an arbitrary vector space is also sometimes referred to as a Hamel basis.

A topological basis (a basis of a topological vector space $ X $ over a field $ K $) is a set $ A = \\{ {a _ {t} } : {t \in T } \\} \subset X $ with properties and functions analogous to those of the algebraic basis of the vector space. The concept of a topological basis, which is one of the most important ones in functional analysis, generalizes the concept of an algebraic basis with regard to the topological structure of $ X $ and makes it possible to obtain, for each element $ X $, its decomposition with respect to the basis $ \\{ a _ {t} \\} $, which is moreover unique, i.e. a representation of $ x $ as a limit (in some sense) of linear combinations of elements $ a _ {t} $:

$$ x = \lim\limits \sum \xi _ {t} (x)a _ {t} , $$

where $ \xi _ {t} (x) $ are linear functionals on $ X $ with values in $ K $, called the components of $ x $ in the basis $ A $, or the coefficients of the decomposition of $ x $ with respect to the basis $ A $. Clearly, for the decomposition of an arbitrary $ x $ to exist, $ A $ must be a complete set in $ X $, and for such a decomposition to be unique (i.e. for the zero element of $ X $ to have all components equal to zero), $ A $ must be a topologically free set in $ X $.

The sense and the practical significance of a topological basis (which will be simply denoted as a "basis" in what follows) is to establish a bijective linear mapping of $ X $, called the basis mapping, $ \Xi $ into some (depending on $ X $) space $ K(T) $ of functions with values in $ K $, defined on a (topological) space $ T $, viz.:

$$ \Xi (x): x \in X \rightarrow \xi _ {x} (t) \in K(T), $$

where $ \xi _ {x} (t) = \xi _ {t} (x) $, so that, symbolically, $ \\{ \xi _ {t} (X) \\} = K(T) $ and $ \\{ \xi _ {x} (T) \\} = X $. Owing to its concrete, effective definition, the structure of $ K(T) $ is simpler and more illustrative than that of the abstractly given $ X $. For instance, an algebraic basis of an infinite-dimensional Banach space is not countable, while in a number of cases, if the concept of a basis is suitably generalized, the cardinality of $ T $ is substantially smaller, and $ K(T) $ simplifies at the same time.

The space $ K(T) $ contains all functions of finite support, and the set of elements of the basis $ \\{ a _ {t} \\} $ is the bijective inverse image of the set of functions $ \\{ \xi _ {t} (s) \\} $ with only one non-zero value which is equal to one:

$$ a _ {t} = \Xi ^ {-1} [ \xi _ {t} (s) ], $$

where $ \xi _ {t} (s) = 1 $ if $ t = s $, and $ \xi _ {t} (s) = 0 $ if $ t \neq s $. In other words, $ a _ {t} $ is the generator of a one-dimensional subspace $ A _ {t} $ which is complementary in $ X $ to the hyperplane defined by the equation $ \xi _ {t} (x) = 0 $.

Thus, the role of the basis $ \\{ a _ {t} \\} $ is to organize, out of the set of components $ \xi _ {t} (x) $ which constitute the image of $ x $ under the basis mapping, a summable (in some sense) set $ \\{ \xi _ {t} (x) a _ {t} \\} $, i.e. a basis "decomposes" a space $ X $ into a (generalized) direct sum of one-dimensional subspaces:

$$ X = \lim\limits \sum \xi _ {t} (X)A _ {t} . $$

A basis is defined in a similar manner in vector spaces with a uniform, limit (pseudo-topological), linear ( $ L $-), proximity, or other complementary structure.

Generalizations of the concept of a basis may be and in fact have been given in various directions. Thus, the introduction of a topology and a measure on $ T $ leads to the concept of the so-called continuous sum of elements from $ X $ and to corresponding integral representations; the decomposition of the space $ X $ into (not necessarily one-dimensional) components is used in the spectral theory of linear operators; the consideration of arbitrary topological algebras over a field $ K $( e.g. algebras of measures on $ T $ with values in $ K $ or even in $ X $, algebras of projection operators, etc.) instead of $ K(T) $ makes it possible to concretize many notions of abstract duality for topological vector spaces and, in particular, to employ the well-developed apparatus of the theory of characters.

A countable basis, which is the most extensively studied and, from the practical point of view, the most important example of a basis, is a sequence $ \\{ a _ {i} \\} $ of elements of a space $ X $ such that each element $ x $ is in unique correspondence with its series expansion with respect to the basis $ \\{ a _ {i} \\} $

$$ \sum \xi _ {i} (x)a _ {i} ,\ \ \xi _ {i} (x) \in K , $$

which (in the topology of $ X $) converges to $ x $. Here, $ T = \mathbf Z $, and there exists a natural order in it. A countable basis is often simply called a "basis" . A weak countable basis is defined in an analogous manner if weak convergence of the expansion is understood. For instance, the functions $ e ^ {ikt } $, $ k \in \mathbf Z $, form a basis in the spaces $ L _ {p} $, $ 1 < p < \infty $( periodic functions absolutely summable of degree $ p $); on the contrary, these functions do not form a basis in the spaces $ L _ {1} $, $ L _ \infty $( measurable functions which almost everywhere coincide with bounded functions) or $ C ^ {1} $( continuous periodic functions). A necessary, but by far not sufficient, condition for the existence of a countable basis is the separability of $ X $( e.g. a countable basis cannot exist in the space of measurable functions on an interval $ [a, b] $ with values in $ \mathbf R $). Moreover, the space $ l _ \infty $ of bounded sequences, not being separable in the topology of $ l _ \infty $, has no countable basis, but the elements $ a _ {i} = \\{ \delta _ {ik } \\} $, where $ \delta _ {ik } = 1 $ if $ i=k $, and $ \delta _ {ik } = 0 $ if $ i \neq k $, form a basis in the weak topology $ \sigma (l _ \infty , l _ {1} ) $. The question of the existence of a countable basis in separable Banach spaces (the basis problem) has been negatively solved [6]. The analogous problem for nuclear spaces also has a negative solution [7].

A countable basis is, however, not always "well-suited" for applications. For example, the components $ \xi _ {t} (x) $ may be discontinuous, the expansion of $ x $ need not converge unconditionally, etc. In this connection one puts restrictions on the basis or introduces generalizations of it.

A basis of countable type is one of the generalizations of the concept of a countable basis in which, although $ T $ is not countable, nevertheless the decomposition of $ x \in X $ with respect to it has a natural definition: the corresponding space $ K(T) $ consists of functions with countable support. For instance, a complete orthonormal set $ \\{ a _ {t} \\} $ in a Hilbert space $ H $ is a basis; if $ x \in H $, then $ \xi _ {t} (x) = \langle x, a _ {t} \rangle $( where $ \langle \cdot , \cdot \rangle $ is the scalar product in $ H $) for all (except possibly a countable set of) indices $ t \in T $, and the series $ \sum \xi _ {t} a _ {t} $ converges to $ x $. The basis mapping is determined by the orthogonal projections onto the closed subspaces generated by the elements $ a _ {t} $. A basis of the space $ AP $ of all complex-valued almost-periodic functions on $ \mathbf R $ consists of the functions $ e ^ {i t \lambda } $; here, $ T = \mathbf R $, $ K(T) $ is the set of countably-valued functions, and the basis mapping is defined by the formula:

$$ \Xi [x( \lambda )] = \ \lim\limits _ {\tau \rightarrow \infty } \ \frac{1}{2 \tau } \int\limits _ {- \tau } ^ { {+ } \tau } x( \lambda )e ^ {it \lambda } d \lambda . $$

An unconditional basis is a countable basis in a space $ X $ such that the decomposition of any element $ x $ converges unconditionally (i.e. the sum of the series does not change if an arbitrary number of its terms is rearranged). For instance, in $ c _ {0} $( sequences converging to zero) and $ l _ {p} $( sequences summable of degree $ p $, $ 1 \leq p < \infty $) the elements $ a _ {i} = \\{ \delta _ {ik } \\} $ form an unconditional basis; in the space $ C[a, b] $ of continuous functions on the interval $ [a, b] $ no basis can be unconditional. An orthonormal countable basis of a Hilbert space is an unconditional basis. A Banach space with an unconditional basis is weakly complete (accordingly, it has a separable dual space) if and only if it contains no subspace isomorphic to $ c _ {0} $( or, correspondingly, $ l _ {1} $).

Two bases $ \\{ a _ {i} \\} $ and $ \\{ b _ {i} \\} $ of the Banach spaces $ X $ and $ Y $, respectively, are said to be equivalent if there exists a bijective linear mapping $ T : a _ {i} \rightarrow b _ {i} $ that can be extended to an isomorphism between $ X $ and $ Y $; these bases are said to be quasi-equivalent if they become equivalent as a result of a certain rearrangement and normalization of the elements of one of them. In each of the spaces, $ l _ {1} , l _ {2} , c _ {0} $ all normalized unconditional bases are equivalent. However, there exist normalized bases not equivalent to orthonormal ones.

A summable basis — a generalization of the concept of an unconditional basis corresponding to a set $ T $ of arbitrary cardinality and becoming identical with it if $ T = \mathbf Z $— is a set $ A = \\{ {a _ {t} } : {t \in T } \\} $ such that for an arbitrary element $ x \in X $ there exists a set of linear combinations (partial sums) of elements from $ A $, which is called a generalized decomposition of $ x $, which is summable to $ x $. This means that for any neighbourhood $ U \subset X $ of zero it is possible to find a finite subset $ A _ {U} \subset A $ such that for any finite set $ A ^ \prime \supset A _ {U} $ the relation

$$ \left ( \sum _ {t \in A ^ \prime } \xi _ {t} a _ {t} - x \right ) \in U, $$

is true, i.e. when the partial sums form a Cauchy system (Cauchy filter). For instance, an arbitrary orthonormal basis of a Hilbert space is a summable basis. A weakly summable basis is defined in a similar way. A totally summable basis is a summable basis such that there exists a bounded set $ B $ for which the set of semi-norms $ \\{ p _ {B} ( \xi _ {t} a _ {t} ) \\} $ is summable. A totally summable basis is at most countable. In a dual nuclear space all weakly summable bases are totally summable.

An absolute basis (absolutely summable basis) is a summable basis of a locally convex space over a normed field such that for any neighbourhood $ U $ of zero and for each $ t \in T $ the family of semi-norms $ \\{ p _ {U} (a _ {t} ) \\} $ is summable. All unconditional countable bases are absolute, i.e. the series $ \sum | \xi _ {i} (x) | p ( a _ {i} ) $ converges for all $ x \in X $ and all continuous semi-norms $ p ( \cdot ) $. Of all Banach spaces only the space $ l _ {1} $ has an absolute countable basis. If a Fréchet space has an absolute basis, all its unconditional bases are absolute. In nuclear Fréchet spaces any countable basis (if it exists) is absolute [13].

A Schauder basis is a basis $ \\{ {a _ {t} } : {t \in T } \\} $ of a space $ X $ such that the basis mapping defined by it is continuous (and is therefore an isomorphism onto some space $ K(T) $), i.e. a basis in which the components $ \xi _ {t} (x) $ for any $ x \in X $ and, in particular, the coefficients of the decomposition of $ x $ with respect to this basis, are continuous functionals on $ X $. This basis was first defined by J. Schauder [5] for the case $ T = \mathbf Z $. The concept of a Schauder basis is the most important of all modifications of the concept of a basis.

A Schauder basis is characterized by the fact that $ \\{ a _ {t} \\} $ and $ \\{ \xi _ {t} \\} $ form a biorthogonal system. Thus, the sequences $ a _ {i} = \\{ \delta _ {ik } \\} $ form countable Schauder bases in the spaces $ c _ {0} $ and $ l _ {p} $, $ p \geq 1 $. A countable Schauder basis forms a Haar system in the space $ C[a, b] $. In complete metric vector spaces (in particular, in Banach spaces) all countable bases are Schauder bases [10]. In Fréchet spaces the concept of a weak basis and a Schauder basis are identical [11]. In barrelled spaces in which there are no linear continuous functionals, there is also no Schauder basis [8]. However, if a weak Schauder basis exists in these spaces, it is an ordinary Schauder basis [9]. A barrelled locally convex space with a countable Schauder basis is reflexive if and only if this basis is at the same time a shrinking set, i.e. if the $ \\{ \xi _ {t} \\} $ corresponding to it will be a basis in the dual space $ X ^ {*} $ and will be boundedly complete, i.e. if the boundedness of the set of partial sums of a series $ \sum _ {i} \xi _ {i} a _ {i} $ implies that this series is convergent [12]. If a Schauder basis is an unconditional basis in a Banach space, then it is a shrinking set (or a boundedly complete set) if and only if $ X $ does not contain subspaces isomorphic to $ l _ {1} $( or, respectively, to $ c _ {0} $).

A Schauder basis in a locally convex space is equicontinuous if for any neighbourhood $ U $ of zero it is possible to find a neighbourhood $ V $ of zero such that

$$ | \xi _ {t} (x) | \ p _ {U} (a _ {t} ) \leq p _ {V} ( x ) $$

for all $ x \in X, t \in T $. All Schauder bases of a barrelled space are equicontinuous, and each complete locally convex space with a countable equicontinuous basis can be identified with some sequence space [15]. An equicontinuous basis of a nuclear space is absolute.

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