Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space.
Vector spaces generalize
Vector spaces are characterized by their
Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.
Algebraic structures |
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Definition and basic properties
In this article, vectors are represented in boldface to distinguish them from scalars.[nb 1][1]
A vector space over a field F is a non-empty set V together with a binary operation and a binary function that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called scalars.[2]
- The binary operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors.
- The binary function, called scalar multiplication,assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av.[nb 2]
To have a vector space, the eight following axioms must be satisfied for every u, v and w in V, and a and b in F.[3]
Axiom | Meaning |
---|---|
Associativity of vector addition |
u + (v + w) = (u + v) + w |
Commutativity of vector addition |
u + v = v + u |
Identity element of vector addition | There exists an element 0 ∈ V, called the zero vector , such that v + 0 = v for all v ∈ V.
|
Inverse elements of vector addition | For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. |
Compatibility of scalar multiplication with field multiplication | a(bv) = (ab)v [nb 3] |
Identity element of scalar multiplication | 1v = v, where 1 denotes the multiplicative identity in F.
|
Distributivity of scalar multiplication with respect to vector addition |
a(u + v) = au + av |
Distributivity of scalar multiplication with respect to field addition | (a + b)v = av + bv |
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space.[4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered. Such a vector space is called an F-vector space or a vector space over F.[5]
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the field F into the endomorphism ring of this group.[6]
Subtraction of two vectors can be defined as
Direct consequences of the axioms include that, for every and one has
- implies or
Even more concisely, a vector space is a module over a field.[7]
Bases, vector coordinates, and subspaces
- Linear combination
- Given a set G of elements of a F-vector space V, a linear combination of elements of G is an element of V of the form where and The scalars are called the coefficients of the linear combination.[8]
- Linear independence
- The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.[9]
- Linear subspace
- A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W.[10] This implies that every linear combination of elements of W belongs to W. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.[11]
The closure property also implies that every intersection of linear subspaces is a linear subspace.[11] - Linear span
- Given a subset G of a vector space V, the linear span or simply the span of G is the smallest linear subspace of V that contains G, in the sense that it is the intersection of all linear subspaces that contain G. The span of G is also the set of all linear combinations of elements of G.
If W is the span of G, one says that G spans or generates W, and that G is aspanning set or a generating set of W.[12] - Basis and dimension
- A subset of a vector space is a basis if its elements are linearly independent and span the vector space.[13] Every vector space has at least one basis, or many in general (see Basis (linear algebra) § Proof that every vector space has a basis).[14] Moreover, all bases of a vector space have the same cardinality, which is called the dimension of the vector space (see Dimension theorem for vector spaces).[15] This is a fundamental property of vector spaces, which is detailed in the remainder of the section.
Bases are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called
Consider a basis of a vector space V of dimension n over a field F. The definition of a basis implies that every may be written
The
A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W.[10] This implies that every linear combination of elements of W belongs to W. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied. The closure property also implies that every intersection of linear subspaces is a linear subspace.[11]
History
Vector spaces stem from
Vectors were reconsidered with the presentation of
In 1857,
An important development of vector spaces is due to the construction of
Examples
Arrows in the plane
The first example of a vector space consists of
The following shows a few examples: if a = 2, the resulting vector aw has the same direction as w, but is stretched to the double length of w (the second image). Equivalently, 2w is the sum w + w. Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the second image).
Ordered pairs of numbers
A second key example of a vector space is provided by pairs of real numbers x and y. The order of the components x and y is significant, so such a pair is also called an ordered pair. Such a pair is written as (x, y). The sum of two such pairs and the multiplication of a pair with a number is defined as follows:[32]
The first example above reduces to this example if an arrow is represented by a pair of
Coordinate space
The simplest example of a vector space over a field F is the field F itself (as it is an abelian group for addition, a part of the requirements to be a field), equipped with its addition (It becomes vector addition.) and multiplication (It becomes scalar multiplication.). More generally, all n-tuples (sequences of length n)
Complex numbers and other field extensions
The set of
More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field F containing a smaller field E is an E-vector space, by the given multiplication and addition operations of F.[34] For example, the complex numbers are a vector space over R, and the field extension is a vector space over Q.
Function spaces
Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function given by
Linear equations
Systems of
where is the matrix containing the coefficients of the given equations, is the vector denotes the
yields where and are arbitrary constants, and is the
Linear maps and matrices
The relation of two vector spaces can be expressed by linear map or linear transformation. They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
An
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see § Examples) are isomorphic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x- and y-component of the arrow, as shown in the image at the right. Conversely, given a pair (x, y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v.[39]
Linear maps V → W between two vector spaces form a vector space HomF(V, W), also denoted L(V, W), or 𝓛(V, W).
Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.[43] If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition.[44] Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F-vector space V is isomorphic to Fn. However, there is no "canonical" or preferred isomorphism; an isomorphism φ : Fn → V is equivalent to the choice of a basis of V, by mapping the standard basis of Fn to V, via φ.
Matrices
Matrices are a useful notion to encode linear maps.[45] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix gives rise to a linear map from Fn to Fm, by the following
Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.[46]
The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.[47] The linear transformation of Rn corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.
Eigenvalues and eigenvectors
Endomorphisms, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.[48] Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to
Basic constructions
In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.
Subspaces and quotient spaces
A nonempty subset of a vector space that is closed under addition and scalar multiplication (and therefore contains the -vector of ) is called a linear subspace of , or simply a subspace of , when the ambient space is unambiguously a vector space.[51][nb 4] Subspaces of are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called its span, and it is the smallest subspace of containing the set . Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of .[52]
Linear subspace of dimension 1 and 2 are referred to as a line, and a plane respectively. If W is an n-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension is called a hyperplane.[53]
The counterpart to subspaces are quotient vector spaces.[54] Given any subspace , the quotient space (" modulo ") is defined as follows: as a set, it consists of
The kernel of a linear map consists of vectors that are mapped to in .[55] The kernel and the image are subspaces of and , respectively.[56]
An important example is the kernel of a linear map for some fixed matrix . The kernel of this map is the subspace of vectors such that , which is precisely the set of solutions to the system of homogeneous linear equations belonging to . This concept also extends to linear differential equations
The existence of kernels and images is part of the statement that the
Direct product and direct sum
The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.
The direct product of a family of vector spaces consists of the set of all tuples , which specify for each index in some index set an element of .[59] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum (also called coproduct and denoted ), where only tuples with finitely many nonzero vectors are allowed. If the index set is finite, the two constructions agree, but in general they are different.
Tensor product
The tensor product or simply of two vector spaces and is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map from the Cartesian product is called bilinear if is linear in both variables and That is to say, for fixed the map is linear in the sense above and likewise for fixed
The tensor product is a particular vector space that is a universal recipient of bilinear maps as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors
Vector spaces with additional structure
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions
A vector space may be given a
Normed vector spaces and inner product spaces
"Measuring" vectors is done by specifying a
Coordinate space can be equipped with the standard dot product:
Topological vector spaces
Convergence questions are treated by considering vector spaces carrying a compatible
In such topological vector spaces one can consider
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any
From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) maps between topological vector spaces are required to be continuous.[72] In particular, the (topological) dual space consists of continuous functionals (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[73]
Banach spaces
Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.[74]
A first example is the vector space consisting of infinite vectors with real entries whose -norm given by
The topologies on the infinite-dimensional space are inequivalent for different For example, the sequence of vectors in which the first components are and the following ones are converges to the
More generally than sequences of real numbers, functions are endowed with a norm that replaces the above sum by the
The space of
These spaces are complete.[75] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.[nb 8]) Concretely this means that for any sequence of Lebesgue-integrable functions with satisfying the condition
Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.[76]
Hilbert spaces
Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.[77] The Hilbert space with inner product given by
By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the
The solutions to various
Algebras over fields
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional
For example, the set of all polynomials forms an algebra known as the
Another crucial example are Lie algebras, which are neither commutative nor associative, but the failure to be so is limited by the constraints ( denotes the product of and ):
- (anticommutativity), and
- (Jacobi identity).[87]
Examples include the vector space of -by- matrices, with the commutator of two matrices, and endowed with the cross product.
The tensor algebra is a formal way of adding products to any vector space to obtain an algebra.[88] As a vector space, it is spanned by symbols, called simple tensors
Related structures
Vector bundles
A vector bundle is a family of vector spaces parametrized continuously by a topological space X.[90] More precisely, a vector bundle over X is a topological space E equipped with a continuous map
Properties of certain vector bundles provide information about the underlying topological space. For example, the
The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.
Modules
Modules are to
Affine and projective spaces
Roughly, affine spaces are vector spaces whose origins are not specified.
The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of
Notes
- ^ It is also common, especially in physics, to denote vectors with an arrow on top: It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.
- scalar product, which is an additional operation on some specific vector spaces, called inner product spaces. Scalar multiplication is the multiplication of a vector by a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar.
- ^ This axiom is not an associative property, since it refers to two different operations, scalar multiplication and field multiplication. So, it is independent from the associativity of field multiplication, which is assumed by field axioms.
- zero vector, while an affine subspace does not necessarily contain it.
- ^ Some authors, such as Roman (2005), choose to start with this equivalence relation and derive the concrete shape of from this.
- uniform structure, Bourbaki (1989), loc = ch. II.
- ^ The triangle inequality for is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm.
- ^ "Many functions in of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", Dudley (1989), §5.3, p. 125.
- ^ For is not a Hilbert space.
- Hamel basis.
- ^ That is, there is a homeomorphism from π−1(U) to V × U which restricts to linear isomorphisms between fibers.
- ^ A line bundle, such as the tangent bundle of S1 is trivial if and only if there is a section that vanishes nowhere, see Husemoller (1994), Corollary 8.3. The sections of the tangent bundle are just vector fields.
Citations
- ^ Lang 2002.
- ^ Brown 1991, p. 86.
- ^ Roman 2005, ch. 1, p. 27.
- ^ Brown 1991, p. 87.
- ^ Springer 2000, p. 185; Brown 1991, p. 86.
- ^ Atiyah & Macdonald 1969, p. 17.
- ^ Bourbaki 1998, §1.1, Definition 2.
- ^ Brown 1991, p. 94.
- ^ Brown 1991, pp. 99–101.
- ^ a b Brown 1991, p. 92.
- ^ a b c Stoll & Wong 1968, p. 14.
- ^ Roman 2005, pp. 41–42.
- ^ Lang 1987, p. 10–11; Anton & Rorres 2010, p. 212.
- ^ Blass 1984.
- ^ Joshi 1989, p. 450.
- ^ Heil 2011, p. 126.
- ^ Halmos 1948, p. 12.
- ^ Bourbaki 1969, ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91.
- ^ Bolzano 1804.
- ^ Möbius 1827.
- ^ Bellavitis 1833.
- ^ Dorier 1995.
- ^ Hamilton 1853.
- ^ Grassmann 2000.
- ^ Peano 1888, ch. IX.
- ^ Guo 2021.
- ^ Moore 1995, pp. 268–271.
- ^ Banach 1922.
- ^ Dorier 1995; Moore 1995.
- ^ Kreyszig 2020, p. 355.
- ^ Kreyszig 2020, p. 358–359.
- ^ Jain 2001, p. 11.
- ^ Lang 1987, ch. I.1.
- ^ Lang 2002, ch. V.1.
- ^ Lang 1993, ch. XII.3., p. 335.
- ^ Lang 1987, ch. VI.3..
- ^ Roman 2005, ch. 2, p. 45.
- ^ Lang 1987, ch. IV.4, Corollary, p. 106.
- ^ Nicholson 2018, ch. 7.3.
- ^ Lang 1987, Example IV.2.6.
- ^ Lang 1987, ch. VI.6.
- ^ Halmos 1974, p. 28, Ex. 9.
- ^ Lang 1987, Theorem IV.2.1, p. 95.
- ^ Roman 2005, Th. 2.5 and 2.6, p. 49.
- ^ Lang 1987, ch. V.1.
- ^ Lang 1987, ch. V.3., Corollary, p. 106.
- ^ Lang 1987, Theorem VII.9.8, p. 198.
- ^ Roman 2005, ch. 8, p. 135–156.
- ^ & Lang 1987, ch. IX.4.
- ^ Roman 2005, ch. 8, p. 140.
- ^ Roman 2005, ch. 1, p. 29.
- ^ Roman 2005, ch. 1, p. 35.
- ^ Nicholson 2018, ch. 10.4.
- ^ Roman 2005, ch. 3, p. 64.
- ^ Lang 1987, ch. IV.3..
- ^ Roman 2005, ch. 2, p. 48.
- ^ Nicholson 2018, ch. 7.4.
- ^ Mac Lane 1998.
- ^ Roman 2005, ch. 1, pp. 31–32.
- ^ Lang 2002, ch. XVI.1.
- ^ Roman (2005), Th. 14.3. See also Yoneda lemma.
- ^ Rudin 1991, p.3.
- ^ Schaefer & Wolff 1999, pp. 204–205.
- ^ Bourbaki 2004, ch. 2, p. 48.
- ^ Roman 2005, ch. 9.
- ^ Naber 2003, ch. 1.2.
- ^ Treves 1967; Bourbaki 1987.
- ^ Schaefer & Wolff 1999, p. 7.
- ^ Kreyszig 1989, §4.11-5
- ^ Kreyszig 1989, §1.5-5
- ^ Choquet 1966, Proposition III.7.2.
- ^ Treves 1967, p. 34–36.
- ^ Lang 1983, Cor. 4.1.2, p. 69.
- ^ Treves 1967, ch. 11.
- ^ Treves 1967, Theorem 11.2, p. 102.
- ^ Evans 1998, ch. 5.
- ^ Treves 1967, ch. 12.
- ^ Dennery & Krzywicki 1996, p.190.
- ^ Lang 1993, Th. XIII.6, p. 349.
- ^ Lang 1993, Th. III.1.1.
- ^ Choquet 1966, Lemma III.16.11.
- ^ Kreyszig 1999, Chapter 11.
- ^ Griffiths 1995, Chapter 1.
- ^ Lang 1993, ch. XVII.3.
- ^ Lang 2002, ch. III.1, p. 121.
- ^ Eisenbud 1995, ch. 1.6.
- ^ Varadarajan 1974.
- ^ Lang 2002, ch. XVI.7.
- ^ Lang 2002, ch. XVI.8.
- ^ Spivak 1999, ch. 3.
- ^ Kreyszig 1991, §34, p. 108.
- ^ Eisenberg & Guy 1979.
- ^ Atiyah 1989.
- ^ Artin 1991, ch. 12.
- ^ Grillet 2007.
- ^ Meyer 2000, Example 5.13.5, p. 436.
- ^ Meyer 2000, Exercise 5.13.15–17, p. 442.
- ^ Coxeter 1987.
References
Algebra
- Anton, Howard; Rorres, Chris (2010), Elementary Linear Algebra: Applications Version (10th ed.), John Wiley & Sons
- ISBN 978-0-89871-510-1
- Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
- Grillet, Pierre Antoine (2007), Abstract algebra, vol. 242, Springer Science & Business Media, ISBN 978-0-387-71568-1
- Halmos, Paul R.(1948), Finite Dimensional Vector Spaces, vol. 7, Princeton University Press
- Heil, Christopher (2011), A Basis Theory Primer: Expanded Edition, Applied and Numerical Harmonic Analysis, Birkhäuser, ISBN 978-0-8176-4687-5
- Jain, M. C. (2001), Vector Spaces and Matrices in Physics, CRC Press, ISBN 978-0-8493-0978-6
- Joshi, K. D. (1989), Foundations of Discrete Mathematics, John Wiley & Sons
- Kreyszig, Erwin (2020), Advanced Engineering Mathematics, John Wiley & Sons, ISBN 978-1-119-45592-9
- Lang, Serge (1987), Linear algebra, Undergraduate Texts in Mathematics (3rd ed.), Springer, ISBN 978-1-4757-1949-9
- MR 1878556
- ISBN 978-0-8218-1646-2
- Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, ISBN 978-0-89871-454-8
- Nicholson, W. Keith (2018), Linear Algebra with Applications, Lyryx
- ISBN 978-0-387-24766-3
- Spindler, Karlheinz (1993), Abstract Algebra with Applications: Volume 1: Vector spaces and groups, CRC, ISBN 978-0-8247-9144-5
- Springer, T.A. (2000), Linear Algebraic Groups, Springer, ISBN 978-0-8176-4840-4
- Stoll, R. R.; Wong, E. T. (1968), Linear Algebra, Academic Press
- ISBN 978-3-540-56799-8
Analysis
- ISBN 978-3-540-13627-9
- ISBN 978-3-540-41129-1
- Braun, Martin (1993), Differential equations and their applications: an introduction to applied mathematics, Berlin, New York: ISBN 978-0-387-97894-9
- BSE-3 (2001) [1994], "Tangent plane", Encyclopedia of Mathematics, EMS Press
- Choquet, Gustave (1966), Topology, Boston, MA: Academic Press
- Dennery, Philippe; Krzywicki, Andre (1996), Mathematics for Physicists, Courier Dover Publications, ISBN 978-0-486-69193-0
- Dudley, Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-10050-6
- Dunham, William (2005), The Calculus Gallery, ISBN 978-0-691-09565-3
- Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: ISBN 978-0-8218-0772-9
- Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, ISBN 978-0-534-17094-3
- Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets, Texts in Applied Mathematics, New York: Springer-Verlag, ISBN 978-0-387-98485-8
- Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001), Digital Signal Processing: A Practical Approach (2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002), ISBN 978-0-201-59619-9
- Krantz, Steven G. (1999), A Panorama of Harmonic Analysis, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America, ISBN 978-0-88385-031-2
- ISBN 978-0-471-85824-9
- MR 0992618
- ISBN 978-0-201-14179-5
- ISBN 978-0-387-94001-4
- Loomis, Lynn H. (2011) [1953], An introduction to abstract harmonic analysis, Dover, OCLC 702357363
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. OCLC 144216834.
- Rudin, Walter (1991), Functional analysis (2 ed.), McGraw-Hill, ISBN 0070542368
- OCLC 840278135.
- Treves, François (1967), Topological vector spaces, distributions and kernels, Boston, MA: Academic Press
Historical references
- ISSN 0016-2736
- Bolzano, Bernard (1804), Betrachtungen über einige Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in German)
- Bellavitis, Giuso (1833), "Sopra alcune applicazioni di un nuovo metodo di geometria analitica", Il poligrafo giornale di scienze, lettre ed arti, 13, Verona: 53–61.
- Bourbaki, Nicolas (1969), Éléments d'histoire des mathématiques (Elements of history of mathematics) (in French), Paris: Hermann
- Dorier, Jean-Luc (1995), "A general outline of the genesis of vector space theory", MR 1347828
- Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur (in French), Chez Firmin Didot, père et fils
- ISBN 978-0-8218-2031-5
- Guo, Hongyu (2021-06-16), What Are Tensors Exactly?, World Scientific, ISBN 978-981-12-4103-1
- Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy
- Möbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German), archived from the original on 2006-11-23
- Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940",
- Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian), Turin
{{citation}}
: CS1 maint: location missing publisher (link) - Peano, G. (1901) Formulario mathematico: vct axioms via Internet Archive
Further references
- ISBN 978-0-03-083993-1
- MR 1043170
- Atiyah, Michael Francis; Macdonald, Ian Grant (1969), Introduction to Commutative Algebra, Advanced Book Classics, Addison-Wesley
- Blass, Andreas (1984), "Existence of bases implies the axiom of choice" (PDF), Axiomatic set theory, Contemporary Mathematics volume 31, Providence, R.I.: MR 0763890
- ISBN 978-3-540-64243-5
- ISBN 978-3-540-64241-1
- ISBN 978-0-387-96532-1
- Eisenberg, Murray; Guy, Robert (1979), "A proof of the hairy ball theorem", JSTOR 2320587
- MR 1322960
- Goldrei, Derek (1996), Classic Set Theory: A guided independent study (1st ed.), London: ISBN 978-0-412-60610-6
- ISBN 978-0-13-124405-4
- ISBN 978-0-387-90093-3
- Halpern, James D. (Jun 1966), "Bases in Vector Spaces and the Axiom of Choice", JSTOR 2035388
- Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013), Calculus : Single and Multivariable (6 ed.), ISBN 978-0470-88861-2
- Husemoller, Dale (1994), Fibre Bundles (3rd ed.), Berlin, New York: ISBN 978-0-387-94087-8
- Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: ISBN 978-3-540-25907-7
- ISBN 978-0-486-66721-8
- Kreyszig, Erwin (1999), Advanced Engineering Mathematics (8th ed.), New York: ISBN 978-0-471-15496-9
- Luenberger, David (1997), Optimization by vector space methods, New York: ISBN 978-0-471-18117-0
- ISBN 978-0-387-98403-2
- ISBN 978-0-7167-0344-0
- Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: MR 2044239
- S2CID 9738629
- Spivak, Michael (1999), A Comprehensive Introduction to Differential Geometry (Volume Two), Houston, TX: Publish or Perish
- ISBN 978-0-412-10800-6
- Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations, ISBN 978-0-13-535732-3
- Wallace, G.K. (Feb 1992), "The JPEG still picture compression standard" (PDF), IEEE Transactions on Consumer Electronics, 38 (1): xviii–xxxiv, ISSN 0098-3063, archived from the original(PDF) on 2007-01-13, retrieved 2017-10-25
- OCLC 36131259.
External links
- "Vector space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]