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.

References

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