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Bilinear map

In mathematics, a bilinear map is a function f: V \times W \to U between vector spaces V, W, and U over a field k that is linear in each argument separately, meaning that for fixed w \in W, the map v \mapsto f(v, w) is linear from V to U, and for fixed v \in V, the map w \mapsto f(v, w) is linear from W to U.^[1]^[2] Bilinear maps generalize linear maps to two inputs and form a fundamental concept in multilinear algebra, where the space of all bilinear maps from V \times W to U itself forms a vector space of dimension (\dim V)(\dim W)(\dim U).^[1] With respect to bases of V, W, and U, any bilinear map can be represented by a collection of coefficients that encode its action on basis elements, analogous to matrix representations for linear maps.^[1] A key property is that bilinear maps preserve addition and scalar multiplication in each variable independently, enabling compositions and universal constructions like tensor products.^[1] A prominent special case occurs when U = k, yielding a bilinear form f: V \times W \to k, which is linear in each argument and often used to define inner products or duality pairings between spaces.^[3] For instance, the standard dot product on \mathbb{R}^n is a bilinear form, and more generally, any bilinear form on a finite-dimensional space admits a matrix representation f(v, w) = v^T A w for some matrix A, with change of basis transforming A via congruence.^[3] Symmetric bilinear forms (where f(v, w) = f(w, v)) are particularly important, as they induce quadratic forms q(v) = f(v, v), which classify geometries like Euclidean or hyperbolic spaces.^[3] Beyond linear algebra, bilinear maps appear in diverse applications, including the tensor product construction, where the universal bilinear map V \times W \to V \otimes W linearizes multilinear expressions.^[1] In algebraic geometry and representation theory, they model pairings between modules, while in cryptography, bilinear pairings—non-degenerate bilinear maps from elliptic curve groups to a multiplicative group—underpin protocols like identity-based encryption and short signatures.^[4]

Definition

Over vector spaces

In the context of vector spaces, a bilinear map is a function B: V \times W \to U, where V, W, and U are vector spaces over the same field K, such that for each fixed element in the second argument, B is linear as a map from V to U, and for each fixed element in the first argument, B is linear as a map from W to U.^[5] This linearity is expressed explicitly by the following conditions: for all v_1, v_2 \in V, w \in W, and \lambda \in K,

\begin{align*} B(v_1 + v_2, w) &= B(v_1, w) + B(v_2, w), \\ B(\lambda v_1, w) &= \lambda B(v_1, w), \end{align*}

and symmetrically for the second argument: for all v \in V, w_1, w_2 \in W, and \lambda \in K,

\begin{align*} B(v, w_1 + w_2) &= B(v, w_1) + B(v, w_2), \\ B(v, \lambda w_2) &= \lambda B(v, w_2). \end{align*}

^[6] These properties ensure additivity and homogeneity with respect to scalar multiplication in each variable separately when the other is held fixed.^[6] The zero map, defined by B(v, w) = 0_U for all v \in V and w \in W, where $0_U is the zero vector in U, satisfies these conditions and thus qualifies as bilinear.^[6] When U = K, bilinear maps are termed bilinear forms; a basic instance is the scalar multiplication map viewed as K \times K \to K, (\lambda, \mu) \mapsto \lambda \mu, which is bilinear since K is a one-dimensional vector space over itself.^[6] The definition of bilinear maps extends naturally to modules over commutative rings, where linearity is replaced by module homomorphisms.^[6]

Over modules

In the more general setting of modules over a commutative ring, the notion of a bilinear map extends the vector space case by replacing field scalars with ring elements. Let R be a commutative ring (typically with identity), and let M, N, and P be R-modules. A map B: M \times N \to P is called an R-bilinear map if it is additive in each argument separately—that is,

B(m_1 + m_2, n) = B(m_1, n) + B(m_2, n), \quad B(m, n_1 + n_2) = B(m, n_1) + B(m, n_2)

for all m_1, m_2 \in M and n_1, n_2 \in N—and homogeneous over R, meaning

B(r m, n) = r B(m, n) = B(m, r n)

for all r \in R, m \in M, and n \in N.^[7]^[8] A key distinction from the vector space setting arises because R need not be a field, so its elements are not necessarily invertible. This lack of inverses means that homogeneity does not allow for division by scalars; for instance, if r B(m, n) = 0 for some nonzero r \in R, it does not imply B(m, n) = 0, reflecting potential torsion in the modules. A concrete illustration occurs when R = \mathbb{Z}, where modules are abelian groups and \mathbb{Z}-bilinear maps are precisely the biadditive maps B: A \times B \to C satisfying B(n a, b) = n B(a, b) = B(a, n b) for integers n, without the ability to "divide" by n unless it is a unit.^[7]^[9] A canonical example of an R-bilinear map is the natural projection \mu: M \times N \to M \otimes_R N, where M \otimes_R N denotes the tensor product module. This map satisfies \mu(m_1 + m_2, n) = \mu(m_1, n) + \mu(m_2, n), \mu(m, n_1 + n_2) = \mu(m, n_1) + \mu(m, n_2), and \mu(r m, n) = r \mu(m, n) = \mu(m, r n), and it is universal in the sense that any other R-bilinear map B: M \times N \to P factors uniquely through \mu via an R-linear map M \otimes_R N \to P. This construction underpins much of module theory over rings.^[7]^[8]

Algebraic properties

Basic properties

A bilinear map B: V \times W \to U between vector spaces over a field K satisfies linearity in each argument separately, fixing the other. Specifically, for all scalars a, b \in K, vectors v_1, v_2 \in V, and w \in W,

B(a v_1 + b v_2, w) = a B(v_1, w) + b B(v_2, w),

and similarly for the second argument with the first fixed.^[7] This implies additivity in each variable: B(v, w_1 + w_2) = B(v, w_1) + B(v, w_2) for v \in V and w_1, w_2 \in W, which follows directly from the linearity in the second argument.^[7] The same holds for homogeneity: B(v, c w) = c B(v, w) for c \in K.^[7] These properties ensure that bilinearity is preserved under precomposition with linear maps. If f: V' \to V and g: W' \to W are linear maps, then the composed map B \circ (f \times g): V' \times W' \to U, defined by (v', w') \mapsto B(f(v'), g(w')), is bilinear.^[1] To verify, linearity in the first argument follows from B(f(a v_1' + b v_2'), g(w')) = B(a f(v_1') + b f(v_2'), g(w')) = a B(f(v_1'), g(w')) + b B(f(v_2'), g(w')), using the linearity of f and B in its first input; the second argument is analogous.^[1] Bilinear maps are a special case of multilinear maps, specifically those of arity two, where multilinearity requires linearity in each of multiple arguments while fixing the others.^[10] Iterating bilinear maps—such as composing a bilinear map with linear maps or using tensor products—yields multilinear maps of higher arity. For instance, the tensor product construction extends bilinearity to produce maps linear in more variables.^[11] The space of bilinear maps \mathrm{Bilin}(V \times W, U) is naturally isomorphic to the space of linear maps \mathrm{Hom}(V \otimes W, U), where V \otimes W is the tensor product of V and W.^[11] This isomorphism arises from the universal property of the tensor product: the canonical bilinear map \otimes: V \times W \to V \otimes W given by (v, w) \mapsto v \otimes w is universal, meaning that for any bilinear map \phi: V \times W \to U, there exists a unique linear map \tilde{\phi}: V \otimes W \to U such that \phi = \tilde{\phi} \circ \otimes, i.e., \phi(v, w) = \tilde{\phi}(v \otimes w).^[11] Conversely, any linear map L: V \otimes W \to U composes with \otimes to yield a bilinear map L \circ \otimes: V \times W \to U.^[1] This bijection provides the explicit isomorphism between the spaces.^[7]

Symmetry and non-degeneracy

A symmetric bilinear map B: V \times W \to K satisfies B(v, w) = B(w, v) for all v \in V, w \in W, assuming V = W for the symmetry to hold in the standard sense.^[12] When the codomain K is the base field and V = W, such a map is termed a symmetric bilinear form.^[12] Non-degeneracy strengthens the properties of a bilinear map B: V \times [W](/page/W) \to K. The map is left non-degenerate if for every nonzero v \in V, there exists w \in [W](/page/W) such that B(v, w) \neq 0; equivalently, the kernel of the left adjoint map \psi_L: V \to W^*, defined by \psi_L(v)(w) = B(v, w), is trivial.^[13] Similarly, it is right non-degenerate if for every nonzero w \in [W](/page/W), there exists v \in V such that B(v, w) \neq 0, corresponding to the trivial kernel of the right adjoint \psi_R: W \to V^*.^[13] The bilinear map is fully non-degenerate if it is both left and right non-degenerate, which, for finite-dimensional spaces with \dim V = \dim [W](/page/W), implies that both adjoint maps are isomorphisms.^[13] Non-degenerate bilinear forms play a central role in duality theory for vector spaces. Specifically, a non-degenerate form B: V \times V \to K induces a linear isomorphism V \to V^* via the map v \mapsto (w \mapsto B(v, w)), identifying the space with its dual and facilitating dual pairings in algebraic structures.^[13]^[14] This isomorphism extends to general non-degenerate pairings between distinct spaces V and W, where B provides a natural duality V \cong W^*.^[12] An important variant is the alternating bilinear form, which is antisymmetric, satisfying B(v, w) = -B(w, v) for all v, w, and thus B(v, v) = 0.^[12] When also non-degenerate, such a form defines a symplectic structure on the space, requiring even dimension over fields of characteristic not 2, and underpins algebraic models of symplectic geometry.^[15]

Examples

Standard algebraic examples

One fundamental example of a bilinear map arises in the construction of the tensor product of two vector spaces over a field K. For vector spaces V and W, the tensor product V \otimes_K W is defined as the quotient of the free vector space on V \times W by the relations enforcing bilinearity, equipped with the natural bilinear map \iota: V \times W \to V \otimes_K W given by (v, w) \mapsto v \otimes w. This map is universal in the sense that for any vector space U and any bilinear map f: V \times W \to U, there exists a unique linear map \tilde{f}: V \otimes_K W \to U such that f = \tilde{f} \circ \iota.^[7] In the context of modules over a commutative ring R, a canonical bilinear pairing is provided by the evaluation map \mathrm{ev}: \mathrm{Hom}_R(M, N) \times M \to N, defined by (\phi, m) \mapsto \phi(m), where \phi \in \mathrm{Hom}_R(M, N) and m \in M. This map is R-bilinear because homomorphisms are linear and the action respects the module structure. It plays a key role in adjointness relations, such as the natural isomorphism \mathrm{Hom}_R(M, \mathrm{Hom}_R(N, P)) \cong \mathrm{Hom}_R(M \otimes_R N, P) for appropriate modules.^[16] Matrix multiplication furnishes another standard algebraic example of bilinearity. Consider the spaces of matrices over a field K: the map \mu: M_{m \times n}(K) \times M_{n \times p}(K) \to M_{m \times p}(K) given by (A, B) \mapsto AB is bilinear, as it is linear in each argument separately when the other is fixed, due to the distributive properties of matrix addition and scalar multiplication. This structure underlies the tensorial view of matrix products and algorithms for their computation.^[17] In associative algebras, the multiplication operation itself defines a bilinear map. An associative algebra A over a field K is a vector space equipped with a bilinear map m: A \times A \to A, (a, b) \mapsto ab, satisfying a(bc) = (ab)c for all a, b, c \in A. If A is unital, there exists an identity element $1 \in A such that $1a = a1 = a. This bilinear multiplication generalizes ring structures to vector spaces and is central to representation theory and algebraic geometry.^[1]

Geometric and analytic examples

In Euclidean space \mathbb{R}^n, the dot product defines a fundamental example of a symmetric bilinear form, given by B(\mathbf{x}, \mathbf{y}) = \mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^n x_i y_i, where \mathbf{x} = (x_1, \dots, x_n) and \mathbf{y} = (y_1, \dots, y_n).^[12] This form is symmetric since B(\mathbf{x}, \mathbf{y}) = B(\mathbf{y}, \mathbf{x}) and non-degenerate, as the only vector orthogonal to all others under this form is the zero vector, enabling the definition of lengths and angles in the space.^[12] On a Riemannian manifold M, the metric tensor g provides a geometric bilinear form at each point p \in M, defined as g_p: T_p M \times T_p M \to \mathbb{R}, where T_p M is the tangent space at p.^[18] This form is symmetric (g_p(u, v) = g_p(v, u)) and positive definite, assigning lengths to tangent vectors via \|u\|_p = \sqrt{g_p(u, u)} and angles between them, thus equipping the manifold with a local Euclidean structure that facilitates measurements of distances and curvatures.^[18] The polarization identity relates quadratic forms to their underlying bilinear forms, allowing recovery of the bilinear structure from the diagonal. For a symmetric bilinear form B over \mathbb{R}, if Q(v) = B(v, v), then B(v, w) = \frac{1}{4} [Q(v + w) - Q(v - w)].^[19] Over \mathbb{C}, the identity extends to B(v, w) = \frac{1}{4} [Q(v + w) - Q(v - w) + i Q(v + i w) - i Q(v - i w)], accommodating the complex sesquilinear case while preserving bilinearity in the first argument and antilinearity in the second.^[12] In probability theory, the covariance function acts as a bilinear form on the space of random variables with finite second moments, particularly for centered variables (mean zero), where B(X, Y) = \mathbb{E}[XY] for random variables X and Y.^[20] This form is symmetric (\mathbb{E}[XY] = \mathbb{E}[YX]) and captures linear dependencies, with the covariance matrix representing it on finite-dimensional spaces of random vectors, enabling analysis of joint variability in stochastic processes.^[20]

Topological aspects

Continuity conditions

In the context of topological vector spaces V, W, and U, a bilinear map B: V \times W \to U is continuous with respect to the product topology on V \times W if the preimage under B of every open set in U is open in V \times W. The product topology is the initial topology making the projections \pi_V: V \times W \to V and \pi_W: V \times W \to W continuous, with a basis consisting of sets O_V \times O_W where O_V is open in V and O_W is open in W.^[21] A key sufficient condition for continuity is that B is continuous at the origin (0,0). Due to the bilinearity of B, which implies B(\lambda v, \mu w) = \lambda \mu B(v, w) for scalars \lambda, \mu and the continuity of scalar multiplication and addition in topological vector spaces, continuity at (0,0) extends to continuity everywhere. This holds in general locally convex spaces, where the topology is defined by a family of seminorms.^[22] In the special case of normed spaces, continuity is equivalent to boundedness: there exists a constant C \geq 0 such that \|B(v, w)\| \leq C \|v\| \|w\| for all v \in V, w \in W. This boundedness condition ensures that B is continuous at (0,0) and hence globally continuous, as the product topology on normed spaces coincides with the topology induced by the product norm.^[23] Separate continuity—meaning B is continuous in each variable when the other is fixed—implies joint continuity under additional assumptions, such as when at least one of V or W is finite-dimensional. In this case, fixing an element in the finite-dimensional space yields a finite-dimensional range for the partial maps, allowing the use of equivalence between separate and joint continuity via boundedness estimates. However, in infinite-dimensional settings without completeness, separate continuity does not guarantee joint continuity. For instance, on the space X of real polynomials on [0,1] equipped with the L^1 norm \|P\|_1 = \int_0^1 |P(t)| \, dt, the bilinear map B(P, Q) = \int_0^1 P(t) Q(t) \, dt: X \times X \to \mathbb{R} is separately continuous, since for fixed P, |B(P, Q)| \leq \|P\|_\infty \|Q\|_1 with \|P\|_\infty < \infty, but not jointly continuous. To see the discontinuity, consider P_n(t) = n^{2/3} t^{n-1}; then \|P_n\|_1 = n^{-1/3} \to 0, yet B(P_n, P_n) = n^{4/3} \int_0^1 t^{2n-2} \, dt = n^{4/3} / (2n-1) \approx (1/2) n^{1/3} \to \infty.^[23]^[24] A bilinear map B: V \times W \to U between topological vector spaces is separately continuous if, for every fixed w \in W, the induced linear map v \mapsto B(v, w) is continuous from V to U, and similarly for every fixed v \in V, the map w \mapsto B(v, w) is continuous from W to U.^[25] In finite-dimensional spaces, separate continuity always implies joint continuity of B. However, in infinite-dimensional settings, separate continuity does not necessarily imply joint continuity. For instance, the pointwise multiplication map C^\infty(\mathbb{R}) \times C^\infty_c(\mathbb{R}) \to C^\infty_c(\mathbb{R}), given by (f, g) \mapsto f g, is separately continuous but not jointly continuous, where C^\infty(\mathbb{R}) is a Fréchet space and C^\infty_c(\mathbb{R}) is a strict inductive limit of Fréchet spaces.^[26] Under additional structural assumptions, separate continuity does imply joint continuity. Specifically, if V and W are barrelled topological vector spaces (such as Fréchet spaces), then a separately continuous bilinear map B: V \times W \to U is jointly continuous. This result follows from applications of the Banach-Steinhaus theorem to the family of linear maps induced by fixing elements in one space.^[25] The space of all separately continuous bilinear maps \mathrm{Bilin}(V \times W, U) can be endowed with a topology, such as the compact-open topology (defined via uniform convergence on compact subsets of V \times W), under which the evaluation map (B, (v, w)) \mapsto B(v, w) is continuous. In this topology, the assignment (V, W, U) \mapsto \mathrm{Bilin}(V \times W, U) behaves continuously with respect to natural morphisms between the spaces.^[27] Discontinuous bilinear functionals arise in non-normable Fréchet spaces. For example, certain bilinear forms on spaces of distributions or test functions fail joint continuity despite satisfying separate continuity, highlighting the role of completeness and barrelledness in ensuring equivalence.^[26]

Relation to multilinear maps

Bilinear as special case

A multilinear map on vector spaces V_1 \times \cdots \times V_k to a vector space W is defined as a function that is linear in each argument when the others are held fixed. The bilinear map represents the special case where k=2, reducing to linearity in each of two arguments separately.^[28]^[29] As a direct instance of multilinear maps, bilinear maps satisfy all associated axioms for the two-variable setting, including additivity and homogeneity in each variable independently—often termed iterated linearity. This inheritance ensures that foundational multilinear properties, such as forming a vector space under pointwise operations, apply without modification to the bilinear context.^[30]^[31] Any bilinear map b: V \times W \to U extends uniquely to a linear map \overline{b}: V \otimes W \to U via the tensor product construction, preserving the original values on pure tensors. This extension leverages the universal property of the tensor product for bilinear maps.^[1]^[32] While higher-degree multilinear maps (for k > 2) admit analogous factorizations through iterated tensor products V_1 \otimes \cdots \otimes V_k, they lack the pairwise simplicity of the bilinear case, where the tensor product directly captures the universality for two factors without requiring multi-step associations.^[1]^[7]

Connections to tensors

The tensor product V \otimes W of vector spaces V and W over a field k is equipped with a universal bilinear map \iota: V \times W \to V \otimes W, defined by the property that for any vector space U and any bilinear map B: V \times W \to U, there exists a unique linear map \hat{B}: V \otimes W \to U such that B = \hat{B} \circ \iota.^[33] This universal property characterizes the tensor product up to unique isomorphism and ensures that every bilinear map factors uniquely through the tensor product.^[33] This factorization induces a natural isomorphism of vector spaces \text{Bilin}(V \times W, U) \cong \Hom_k(V \otimes W, U), where \text{Bilin}(V \times W, U) denotes the space of k-bilinear maps from V \times W to U, and \Hom_k denotes the space of k-linear maps. The explicit correspondence sends a bilinear map B to the induced linear map \hat{B} defined by \hat{B}(v \otimes w) = B(v, w) on elementary tensors, with linearity extending to the whole space. In the category of vector spaces \Vect_k, this isomorphism reflects the tensor product serving as the representing object for the functor of bilinear maps. More generally, in the category of modules over a commutative ring R, the tensor product M \otimes_R N satisfies an analogous universal property with respect to R-bilinear maps, and the tensor-hom adjunction provides a natural isomorphism \Hom_R(M \otimes_R N, P) \cong \Hom_R(N, \Hom_R(M, P)) for any R-modules M, N, P.^[34] This adjunction, with -\otimes_R N left adjoint to \Hom_R(N, -), underscores the categorical role of the tensor product in linearizing bilinear maps across module categories.^[34] Iterated applications of the tensor product yield higher-order tensors, where the k-fold tensor product V^{\otimes k} = V \otimes \cdots \otimes V ( k times) universalizes k-linear maps via a multilinear map V^k \to V^{\otimes k}.^[35] For any vector space U and k-linear map f: V^k \to U, there exists a unique linear map \hat{f}: V^{\otimes k} \to U such that f(v_1, \dots, v_k) = \hat{f}(v_1 \otimes \cdots \otimes v_k).^[35] This construction extends the bilinear case, associating multilinear functionals directly to linear functionals on iterated tensors.^[35]

References

[1]
[PDF] multilinear algebra notes - Eugene Lerman
Definition 1.11. An algebra over R is a vector space A together with a bilinear map A×A → A, (a, a0) 7→ aa0 (“multiplication”). An algebra A is said to be an ...
[2]
[PDF] Second Derivatives, Bilinear Maps, and Hessian Matrices
Definition 56 (Bilinear Map). Let U,V,W be a vector spaces, not necessarily the same. Then, a bilinear map is a function B : U ×V → W, mapping a u ∈ U and ...
[3]
[PDF] Further linear algebra. Chapter V. Bilinear and quadratic forms.
Definition 1.1 Let V be a vector space over k. A bilinear form on V is a function f : V × V → k such that. • f(u + λv, w) = f(u, w) + λf(v, w);.Missing: paper | Show results with:paper
[4]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · Let 𝑉 and 𝑊 be vector spaces and 𝐴∶ 𝑉 → 𝑊, a linear map. ... is also a bilinear map of 𝑉1 × 𝑉2 into 𝑊, so the set of all bilinear maps ...
[5]
[PDF] bilinear forms - keith conrad
Letting n = p + q, Rp,q equals Rn as vector spaces, so their dual spaces are the same. ... A bilinear map B: V ×W → F is called perfect if the induced ...
[6]
[PDF] TENSOR PRODUCTS 1. Introduction Let R be a commutative ring ...
Let R be a commutative ring and M and N be R-modules. (We always work with rings having a multiplicative identity and modules are assumed to be unital: 1 ...
[7]
[PDF] Algebraic Topology I: Lecture 20 Tensor Product
Let's begin by recalling the definition of a bilinear map over a commutative ring R. Definition 20.1. Given three R-modules, M, N, P , a bilinear map (or, to be ...
[8]
[PDF] Notes on tensor products Robert Harron - Department of Mathematics
In particular, if R is a commutative ring, M and N are two left R-modules, and we view M with its standard bimodule structure, then M ⊗R N is a left R-module ...<|control11|><|separator|>
[9]
A pairing in homology and the category of linear complexes of tilting ...
A generalization of the bilinear pairing to higher Ext's. 5. Graded non-degeneracy in a graded case. 6. The category of linear complexes of tilting modules. 7 ...
[10]
[PDF] SOME MULTILINEAR ALGEBRA 1. Tensor products. Let V,W be ...
Tensor products have the following useful property: if T : V ×W → E is a bilinear map to a vector space E (that is, linear in each argument with the other ...
[11]
[PDF] Notes on Tensor Products - Brown Math
May 3, 2014 · Existence of the Universal Property: The tensor product has what is ... Lemma 3.1 Suppose that φ : M × N → P is a bilinear map. Then there ...
[12]
[PDF] BILINEAR FORMS The geometry of Rn is controlled algebraically by ...
The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study ...
[13]
[PDF] Linear Algebra: Non-degenerate Bilinear Forms - DPMMS
In these notes we generalise some of the results of that section to non-degenerate forms. 1. Non-degeneracy. Let V and W be finite dimensional vector spaces ...Missing: duality | Show results with:duality
[14]
[PDF] 25. Duals, naturality, bilinear forms
F = D ◦ D associating to vector spaces V their double duals V ∗∗ also gives maps ... k-vectorspace with a k-bilinear map [, ] (the Lie bracket) such that [x, y] ...
[15]
[PDF] 1. Linear algebra preliminaries 1.1. Some facts about bilinear forms ...
Definition. A bilinear form ψ : V ×V → k on a vector space V is called symplectic if it is skew-symmetric and non-degenerate. Furthermore, the pair (V,ψ) ...Missing: duality | Show results with:duality<|control11|><|separator|>
[16]
[PDF] DUAL MODULES 1. Introduction Let R be a commutative ring. For ...
The dual module of M, denoted as M∨, is HomR(M,R), where R-linear maps from M to R are considered. Elements are called linear functionals.
[17]
[PDF] Lecture 9: Algorithms for Matrix Multiplication Part I 1 Tensors and ...
Nov 16, 2021 · It corresponds to a bilinear map Fac × Fbc → Fab. Note that if we plug in x and y from the matrix entries then the coefficient of zi,j ...
[18]
[PDF] Quick Introduction to Riemannian geometry - Arizona Math
In particular, we have the. Riemannian metric tensor. Definition 1 A Riemannian metric g is a two-tensor (i.e., a section of T M. T M) which is symmetric ...
[19]
[PDF] Bilinear and Quadratic Forms This handout should be read just ...
A bilinear form on V is a function B: V × V → R that is linear in each variable separately. It is symmetric if B(v, w) = B(w, v).
[20]
[PDF] Correlation Angles and Inner Products
Aug 9, 2011 · Covariance is used as an inner product on a formal vector space built on n random variables to define measures of correlation Md across a set ...
[21]
[PDF] FUNCTIONAL ANALYSIS | Second Edition Walter Rudin
... Bilinear mappings. 52. Exercises. 53. 3 Convexity. 56. The Hahn-Banach theorems. 56 ... continuity theorem. Closed subspaces of If-spaces. The range of a vector ...
[22]
https://link.springer.com/content/pdf/10.1007/BF01138506.pdf
[23]
[PDF] Continuity of Bilinear Mappings
It is well-known that a linear mapping T : X → Z between two normed linear spaces is continuous if and only if it is continuous at 0 if and only if it is ...Missing: counterexample jointly l2<|separator|>
[24]
[PDF] Functional Analysis (WS 20/21), Problem Set 3 (Banach-Steinhaus ...
Nov 19, 2020 · ... separately continuous ... may be useful. B5. Let X be the space of polynomials P[0,1] equipped with the L1(0,1) norm. We define a bilinear map: ...
[25]
[PDF] arXiv:math/0501187v1 [math.FA] 12 Jan 2005
Jan 12, 2005 · Moreover, separate continuity is equivalent to continuity for bilinear mappings defined on Fréchet or barrelled DF spaces (see [7], Theorem ...
[26]
[PDF] arXiv:0803.1529v2 [math.QA] 10 Jan 2009
Jan 10, 2009 · [M1] Suppose an LF-space A has an associative multiplication that is separately continuous. Separate continuity on a Fréchet space implies joint ...
[27]
[PDF] Continuity of bilinear maps on direct sums of topological vector spaces
Dec 21, 2011 · Abstract. We prove a criterion for continuity of bilinear maps on countable di- rect sums of topological vector spaces.
[28]
[PDF] Appendix 2 - UPenn CIS
× Em → F is a multilinear map (or an m-linear map), iff it is linear in each argument, holding the others fixed. ... which is precisely the bilinear map h1(v1,v2) ...
[29]
[PDF] Lecture 19 Differentiable Manifolds 10/05/2011 Multilinear maps ...
Oct 5, 2011 · Remark 19.2. A multilinear map f : V1 × V2 → U is called a bilinear map. Remark 19.3. In this class we will mostly consider vector spaces ...
[30]
[PDF] Chapter 9 Multilinear Algebra - LSU Math
... vector spaces. Further sup- pose that fi : Vi → Wi is a linear map for each ... The usual dot product in Rn is a bilinear map. In fact, any bilinear ...<|control11|><|separator|>
[31]
[PDF] Multilinear Algebra - Alexander Rhys Duncan
Jan 23, 2023 · As in the bilinear case, multilinear maps form a vector space and the symmetric, skew-symmetric, and alternating forms are subspaces of the ...
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[PDF] Multilinear Mappings and Tensors - UCSD CSE
In this chapter we generalize our earlier discussion of bilinear forms, which leads in a natural manner to the concepts of tensors and tensor products.
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[PDF] Lecture 16: Tensor Products
Let U and V be vector spaces. Then the tensor product is vector space U ⊗ V such that. (1) There is a bilinear map i: U × V → ...
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[PDF] Math200b, lecture 13 - UCSD Math
(a) (m,n) 7→ m ⊗ n is a map from M × N to M ⊗B N that is. B-balanced, A-linear in M, and linear in N. (b) (Tensor-Hom adjunction) There is a natural isomorphism.
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[PDF] Multilinear Algebra and Tensor Symmetries
Aug 28, 2011 · Iterated Tensor Products: Linearizing Multilinear Maps. Associativity of Tensor Product: U, V, W vector spaces. Define bilinear map τ : (U ...