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In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space.When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

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Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces.When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.

Early terms for dual include polarer Raum [Hahn 1927], espace conjugu, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.[1]

If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.

where the bracket [,] on the left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose,[11] and is formally similar to the definition of the adjoint.

The question is: Does the double dual space $\mathcal{D}''(R)$, i.e., the dual space of the distributions $\mathcal{D}'(R)$, exist? For Banach space $X$, we know that double dual $X^{**}\supseteq X$. They are equal when $X$ is reflective. A first guess is that $\mathcal{D}''(R)\supseteq \mathcal{D}(R)$.

Well, that depends on what topology you want to put on the space of distributions. The weak$^*$ is probably not really the one you would like to take. Instead, the strong dual might be more useful. The seminorms of this topology are given by$$p_B(\varphi) = \sup_{f \in B} |\varphi(f)|$$ where $B \subseteq \mathcal{D}(\mathbb{R})$ runs through the bounded subsets of the LF space $\mathcal{D}(\mathbb{R})$. The it is a theorem that (since the test functions are Montel etc) the dual with respect to this is again the space of test functions, i.e. the test functions are reflexive...

I guess for the weak$^\ast$ topology this is not true and one gets a different dual of the dual. Your inclusion is correct, any test function gives a linear functional on the distribtions (by evaluation) which is continuous in the weak$^*$-topology. But you probably get more...

It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of Radon measures with bounded variation.

What you state in the first paragraph is the Riesz Representation Theorem (see wikipedia). This is valid for all locally compact Hausdorff spaces; so in particular for $\mathbb R$ (ah, I guess, if you look at $C_0(\mathbb R)^*$).

If $X$ is any topological space, then of course we can talk of $C^b(X)$ (the bounded continuous functions on $X$). This is still a commutative C$^*$-algebra, and so is isomorphic to $C(K)$, where $K$ is some compact Hausdorff space. The process of moving from $X$ to $K$ is functorial; purely at the topological level it corresponds to constructing the Stone-Cech compactification (see wikipedia). Point evaluation at $x\in X$ induces a character on $C^b(X) = C(K)$ and hence a point $k$ of $K$; we thus get a (continuous) map $X\rightarrow K$. This is injective if $X$ is completely regular; but it can fail to be injective (basically, we might lack enough continuous functions to separate points of $X$).

The problem of obtaining a useful generalisation of the Riesz representation theorem for non-compact spaces was addressed in the 50's by R.C. Buck, amongst others. It was clear that it was necessary to leave the context of Banach spaces for a nice theory. Buck introduced the so-called strict topology on the space of bounded, continuous functions on a locally compact space and showed that the dual is the space of bounded Radon measurs on the underlying space. This was generalised to the case of completely regular spaces in the 60's using the theory of mixed topologies or Saks spaces which had been developed by the Polish school. The most succinct definition of the resulting topology on the above space is that it is the finest locally convex topology which agrees with compact convergence on bounded sets. There is a relatively complete theory---in particular, the Riesz representation theorem holds in its natural form.

We present pseudopotential coefficients for the first two rows of the Periodic Table. The pseudopotential is of an analytic form that gives optimal efficiency in numerical calculations using plane waves as a basis set. At most, seven coefficients are necessary to specify its analytic form. It is separable and has optimal decay properties in both real and Fourier space. Because of this property, the application of the nonlocal part of the pseudopotential to a wave function can be done efficiently on a grid in real space. Real space integration is much faster for large systems than ordinary multiplication in Fourier space, since it shows only quadratic scaling with respect to the size of the system. We systematically verify the high accuracy of these pseudopotentials by extensive atomic and molecular test calculations. 1996 The American Physical Society.

We generalize the concept of separable dual-space Gaussian pseudopotentials to the relativistic case. This allows us to construct this type of pseudopotential for the whole Periodic Table, and we present a complete table of pseudopotential parameters for all the elements from H to Rn. The relativistic version of this pseudopotential retains all the advantages of its nonrelativistic version. It is separable by construction, it is optimal for integration on a real-space grid, it is highly accurate, and, due to its analytic form, it can be specified by a very small number of parameters. The accuracy of the pseudopotential is illustrated by an extensive series of molecular calculations.

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Dual Space Pro - Multi Accounts is a free utility application for mobile by developer Dualspace. It's an app-cloning tool that lets users create multiple instances of their favorite apps. It allows them to log in to multiple accounts at the same time and just from a single device.

This app will work on all kinds of social media platforms, such as Twitter and Snapchat, or even messaging services like Line. You can have two accounts running simultaneously, and it will work with dual instances of the same app or with different ones. You're basically cloning applications, but you can still seamlessly switch between them through your device's native multitasking window function.

Not really picky if it uses virtualization like dual space or alters and APK so it's not recognized as the same app. I've seen both of these methods used- although the later I haven't seen much success at least in the Play store.

Given a vector space $V$, we define its dual space $V^*$ to be the set of all linear transformations $\varphi: V \to \mathbb{F}$. The $\varphi$ is called a linear functional. In other words, $\varphi$ is something that accepts a vector $v \in V$ as input and spits out an element of $\mathbb{F}$ (lets just assume that $\mathbb{F} = \mathbb{R}$, meaning that it spits out a real number). If you take all the possible (linear) ways that a $\varphi$ can eat such vectors and produce real numbers, you get $V^*$.

Recall that the dual of space is a vector space on its own right, since the linear functionals $\varphi$ satisfy the axioms of a vector space. But if $V^*$ is a vector space, then it is perfectly legitimate to think of its dual space, just like we do with any other vector space. This might feel too recursive, but hold on. The double dual space is $(V^*)^* = V^{**}$ and is the set of all linear transformations $\varphi: V^* \to \mathbb{F}$.

When we defined $V^*$ from $V$ we did so by picking a special basis (the dual basis), therefore the isomorphism from $V$ to $V^*$ is not canonical. It turns out that the isomorphism between the initial vetor space $V$ and its double dual, $V^{**}$, is canonical as we shall see right away. Let $v \in V, \varphi \in V^*$ and $\hat{v} \in V^{**}$. We can now define a linear map:

This also happens to explain intuitively some facts. For instance, the fact that there is no canonical isomorphism between a vectorspace and its dual can then be seen as a consequence of the fact that rulers need scaling, and there is no canonical way to provideone scaling for space. However, if we were to measure the measure-instruments, how could we proceed? Is there a canonical way to doso? Well, if we want to measure our measures, why not measure them by how they act on what they are supposed to measure? We need nobases for that. This justifies intuitively why there is a natural embedding of the space on its bidual. (Note, however, that thisfails to justify why it is an isomorphism in the finite-dimensional case). ff782bc1db