Basis

of a set $X$

A minimal subset $B$ that generates it. Generation here means that by application of operations of a certain class $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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

$$n=a_1+\cdots +a_k,$$

where $a_i\in \mathrm{\Omega }$. 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 $\mathrm{\Omega }$ 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 $s\to a=\vee$( which is equivalent to $s\subset a$), where $s\in S$, $a\in \mathfrak{A}$, $\vee$ is the unit of $\mathfrak{A}$ and "" is the operation of implication. One also introduces in an analogous manner a basis for a filter $\mathrm{\nabla }$ as a set $S$ such that for an arbitrary $a\in \mathrm{\nabla }$ 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 $\theta :X\to Y$ 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

$$\mathrm{\Xi }:x\to {\xi }_x(t),$$

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 $\mathrm{\Xi }(X)$ 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 ${\mathrm{\Lambda }}_n$ of this function belongs to the linear envelope of $\beta$. The elements of $\beta$ may be so chosen that they belong to a sequence $\{{\mathrm{\Lambda }}_i\}$; 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 $a_t$:

$$x=lim\sum {\xi }_t(x)a_t,$$

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, $\mathrm{\Xi }$ into some (depending on $X$) space $K(T)$ of functions with values in $K$, defined on a (topological) space $T$, viz.:

$$\mathrm{\Xi }(x):x\in X\to {\xi }_x(t)\in K(T),$$

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:

$$a_t={\mathrm{\Xi }}^{-1}[{\xi }_t(s)],$$

where ${\xi }_t(s)=1$ if $t=s$, and ${\xi }_t(s)=0$ if $t\ne s$. 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:

$$X=lim\sum {\xi }_t(X)A_t.$$

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 $L_p$, $1<p<\mathrm{\infty }$( periodic functions absolutely summable of degree $p$); on the contrary, these functions do not form a basis in the spaces $L_1$, $L_{\mathrm{\infty }}$( 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 $l_{\mathrm{\infty }}$ of bounded sequences, not being separable in the topology of $l_{\mathrm{\infty }}$, has no countable basis, but the elements $a_i=\{{\delta }_{ik}\}$, where ${\delta }_{ik}=1$ if $i=k$, and ${\delta }_{ik}=0$ if $i\ne k$, form a basis in the weak topology $\sigma (l_{\mathrm{\infty }},l_1)$. 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 ${\xi }_t(x)=⟨x,a_t⟩$( where $⟨\cdot ,\cdot ⟩$ 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 $e^{it\lambda }$; here, $T=\mathbf{R}$, $K(T)$ is the set of countably-valued functions, and the basis mapping is defined by the formula:

$$\mathrm{\Xi }[x(\lambda )]=\underset{\tau \to \mathrm{\infty }}{lim}\frac{1}{2\tau }\underset{-\tau }{\overset{+\tau }{\int }}x(\lambda )e^{it\lambda }d\lambda .$$

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 $p$, $1\le p<\mathrm{\infty }$) 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 $T:a_i\to b_i$ 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

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

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 $l_p$, $p\ge 1$. 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 $X^{\ast }$ 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

$$|{\xi }_t(x)|p_U(a_t)\le p_V(x)$$

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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