VI. Weak topologies
... (3) Let Γ be a noenmpty set and let the space FΓ be equipped by the product topology (cf. Example V.1(2)). The product topology equals σ(FΓ , M ) where M = {x 7→ x(γ); γ ∈ Γ}. (4) Let T be a topological space and let C(T, F) be the vector space of all the continuous functions on T . For t ∈ T define ...
... (3) Let Γ be a noenmpty set and let the space FΓ be equipped by the product topology (cf. Example V.1(2)). The product topology equals σ(FΓ , M ) where M = {x 7→ x(γ); γ ∈ Γ}. (4) Let T be a topological space and let C(T, F) be the vector space of all the continuous functions on T . For t ∈ T define ...
VECTOR SUPERIOR AND INFERIOR Y. Chiang In this paper, all
... (ii) If K is bounded above with respect to C, and if there is an x 0 ∈ K such that the set ϕ(K ∩ (x 0 + C)) is closed in IR, then Sup (K , C) is nonempty. Theorem 3.4. Let (Z , C) be an ordered Hausdorff topological vector space, let ϕ : Z −→ IR be the linear functional given in Proposition 3.1, let ...
... (ii) If K is bounded above with respect to C, and if there is an x 0 ∈ K such that the set ϕ(K ∩ (x 0 + C)) is closed in IR, then Sup (K , C) is nonempty. Theorem 3.4. Let (Z , C) be an ordered Hausdorff topological vector space, let ϕ : Z −→ IR be the linear functional given in Proposition 3.1, let ...
Contents - POSTECH Math
... (2) Consider Rn as an abelian topological group under vector addition. The Zn is a normal sub group of Rn , so that Tn ≡ Rn /Zn ≃ S1 × · · · × S1 is the n-torus. Definition 1.5 Let M1 and M2 be 2-manifolds. Let Di ⊆ Mi be imbedded disks, and Mi′ = Mi −int(Di ), i = 1, 2. Then, ∂Mi′ in Mi are homeomo ...
... (2) Consider Rn as an abelian topological group under vector addition. The Zn is a normal sub group of Rn , so that Tn ≡ Rn /Zn ≃ S1 × · · · × S1 is the n-torus. Definition 1.5 Let M1 and M2 be 2-manifolds. Let Di ⊆ Mi be imbedded disks, and Mi′ = Mi −int(Di ), i = 1, 2. Then, ∂Mi′ in Mi are homeomo ...
Package `LSAfun`
... Offers methods and functions for working with Vector Space Models of semantics, such as Latent Semantic Analysis (LSA). Such models are created by algorithms working on a corpus of text documents. Those algorithms achieve a high-dimensional vector representation for word (and document) meanings. The ...
... Offers methods and functions for working with Vector Space Models of semantics, such as Latent Semantic Analysis (LSA). Such models are created by algorithms working on a corpus of text documents. Those algorithms achieve a high-dimensional vector representation for word (and document) meanings. The ...
Note on fiber bundles and vector bundles
... In the diagram π1 : U × F → U is projection onto the first factor. (We will often use the notation πk : X1 × X2 × · · · × Xn → Xk for projection onto the k th factor of a Cartesian product.) The commutation of the diagram is the assertion that π = π1 ◦ ϕ, which means that ϕ maps fibers of π diffeomo ...
... In the diagram π1 : U × F → U is projection onto the first factor. (We will often use the notation πk : X1 × X2 × · · · × Xn → Xk for projection onto the k th factor of a Cartesian product.) The commutation of the diagram is the assertion that π = π1 ◦ ϕ, which means that ϕ maps fibers of π diffeomo ...
Projective limits of topological vector spaces
... Since φi : XI → Xi , i ∈ I, are compatible with the projective system (Xi , φij , I), it follows that φj : XI → Xj , j ∈ J, are compatible with (Xi , φij , J). As (XJ , φ0i , J) is a projective limit of (Xi , φij , I), there is a unique morphism φ : XI → XJ such that φ0j ◦ φ = φj for all j ∈ J. On ...
... Since φi : XI → Xi , i ∈ I, are compatible with the projective system (Xi , φij , I), it follows that φj : XI → Xj , j ∈ J, are compatible with (Xi , φij , J). As (XJ , φ0i , J) is a projective limit of (Xi , φij , I), there is a unique morphism φ : XI → XJ such that φ0j ◦ φ = φj for all j ∈ J. On ...
Differential geometry for physicists
... of the two-dimensional sphere S 2 by latitude −π/2 < θ < π/2 and longitude 0 < ϕ < 2π, which does not include the poles and the zero meridian. It is also conventional to “cure” this problem by redefining the coordinate range in the form −π/2 ≤ θ ≤ π/2 and 0 ≤ ϕ < 2π, but this is not even a chart any ...
... of the two-dimensional sphere S 2 by latitude −π/2 < θ < π/2 and longitude 0 < ϕ < 2π, which does not include the poles and the zero meridian. It is also conventional to “cure” this problem by redefining the coordinate range in the form −π/2 ≤ θ ≤ π/2 and 0 ≤ ϕ < 2π, but this is not even a chart any ...
Background notes
... In the diagram π1 : U × F → U is projection onto the first factor. (We will often use the notation πk : X1 × X2 × · · · × Xn → Xk for projection onto the kth factor of a Cartesian product.) The commutation of the diagram is the assertion that π = π1 ◦ ϕ, which means that ϕ maps fibers of π diffeomor ...
... In the diagram π1 : U × F → U is projection onto the first factor. (We will often use the notation πk : X1 × X2 × · · · × Xn → Xk for projection onto the kth factor of a Cartesian product.) The commutation of the diagram is the assertion that π = π1 ◦ ϕ, which means that ϕ maps fibers of π diffeomor ...
Basic Differentiable Calculus Review
... Definition 2.2 Let E, F be normed vector spaces. A function L : E → F is called a linear operator if (i) for all x, y ∈ E, L(x + y) = L(x) + L(y), and (ii) for all α ∈ F and x ∈ E, L(αx) = αL(x). Proposition 2.3 Suppose that L is a linear operator from the normed vector space E to the normed vector ...
... Definition 2.2 Let E, F be normed vector spaces. A function L : E → F is called a linear operator if (i) for all x, y ∈ E, L(x + y) = L(x) + L(y), and (ii) for all α ∈ F and x ∈ E, L(αx) = αL(x). Proposition 2.3 Suppose that L is a linear operator from the normed vector space E to the normed vector ...
Terse Notes on Riemannian Geometry
... Remember that continuity of a function on the real line is phrased in terms of open intervals, i.e., the usual -δ definition. A topology defines which subsets of a set X are “open”, much in the same way an interval is open. As will be seen at the end of this subsection, open sets in Rn are made up ...
... Remember that continuity of a function on the real line is phrased in terms of open intervals, i.e., the usual -δ definition. A topology defines which subsets of a set X are “open”, much in the same way an interval is open. As will be seen at the end of this subsection, open sets in Rn are made up ...
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ
... (G, G+ ) is an ordered n-dimensional Q-vector space with Riesz interpolation then it is an inductive limit of (Zn , Zn≥0 ) (where the maps are, of course, not unimodular). In [6], David Handelman shows that every vector space with Riesz interpolation can be realized as an inductive limit of ordered ...
... (G, G+ ) is an ordered n-dimensional Q-vector space with Riesz interpolation then it is an inductive limit of (Zn , Zn≥0 ) (where the maps are, of course, not unimodular). In [6], David Handelman shows that every vector space with Riesz interpolation can be realized as an inductive limit of ordered ...
Lecture 1: Connections on principal fibre bundles
... A global section would correspond to choosing a square-root function smoothly on the unit circle. This does not exist, however, since any definition of z 1/2 always has a branch cut from the origin out to the point at infinity. Therefore the bundle is not trivial. In fact, if the bundle were trivial ...
... A global section would correspond to choosing a square-root function smoothly on the unit circle. This does not exist, however, since any definition of z 1/2 always has a branch cut from the origin out to the point at infinity. Therefore the bundle is not trivial. In fact, if the bundle were trivial ...
- Journal of Linear and Topological Algebra
... Tx (y) = y + x. By definition of s-topological vector spaces, there exist semi-open neighbourhoods U and V containing y and x respectively, such that U + V ⊆ W . In particular, we have U + x ⊆ W which means Tx (U ) ⊆ W . The inclusion shows that Tx is semi-continuous at y. Hence Tx is semi-continuou ...
... Tx (y) = y + x. By definition of s-topological vector spaces, there exist semi-open neighbourhoods U and V containing y and x respectively, such that U + V ⊆ W . In particular, we have U + x ⊆ W which means Tx (U ) ⊆ W . The inclusion shows that Tx is semi-continuous at y. Hence Tx is semi-continuou ...
Vector field
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.