What is Dual spaces: Definition and 20 Discussions

In mathematics, any vector space


{\displaystyle V}
has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on


{\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.
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 conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.

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  1. cianfa72

    I Terminologies used to describe tensor product of vector spaces

    Hi, I'm in trouble with the different terminologies used for tensor product of two vectors. Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##. From a...
  2. cianfa72

    I Inner product vs dot/scalar product

    Hi, from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things. Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
  3. L

    I Anyone knows why musical isomorphism is called so?

    Anyone knows why musical isomorphism is called so? Why is it musical? https://en.wikipedia.org/wiki/Musical_isomorphism
  4. patric44

    Why the bra vector is said to belong in the dual space?

    hi i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...
  5. Math Amateur

    MHB Dual Spaces .... Friedberg et al, Example 4, Section 2.6

    I am reading the book: "Linear Algebra" by Stephen Friedberg, Arnold Insel, and Lawrence Spence ... and am currently focused on Section 2.6: Dual Spaces ... ... I need help with an aspect of Example 4, Section 2.6 ... Example 4, Section 2.6 reads as follows: (see below for details of Section...
  6. F

    Prove that three functions form a dual basis

    Homework Statement Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.
  7. D

    What is the meaning of the term "dual"?

    Apologies if this is a really trivial question, but I've never been quite sure as to the usage of the terminology dual space. I get that given a vector space ##V## we can construct a set of linear functionals that map ##V## into its underlying field and that these linear functionals themselves...
  8. D

    Dual spaces and the meaning of "Dual"

    So, I understand that the dual space, V^{\ast} of a vector space V over a scalar field \mathbb{F} is the set of all linear functionals f^{\ast}:V\rightarrow\mathbb{F} that map the vectors in V to the scalar field \mathbb{F}, but I'm confused as to the meaning of the term dual?! Is it just that...
  9. caffeinemachine

    MHB Natural Isomorphism b/w Dual Spaces Tensor Prod & Multilinear Form Space

    I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...
  10. D

    Dual vector spaces and linear maps

    Hi all. I was hoping I could clarify my understanding on some basic notions of dual spaces. Suppose I have a vector space V along with a basis \lbrace\mathbf{e}_{i}\rbrace, then there is a unique linear map \tilde{e}^{i}: V\rightarrow \mathbb{F} defined by \tilde{e}^{i}(\mathbf{v})=v^{i}...
  11. J

    Duality and Orthogonality: What's the Difference?

    1. I cannot understand the difference between orthogonality and duality? Of course orthogonal vectors have dot product zero but isn't this the condition of duality as well? Swinging my head around it my cannot find the answer on the internet as well. 2.Relating to same concept is orthogonality...
  12. T

    Exercise on dual / second dual spaces (functional analysis)

    Homework Statement Let (X,\|\cdot\|) be a reflexive Banach space. Let \{T_n\}_{n\in\mathbb{N}} be a sequence of bounded linear operators from X into X such that \lim_{n\to\infty}f(T_nx) exists for all f\in X' and x\in X. Use the Uniform Boundedness Principle (twice) to show that...
  13. T

    Functional analysis - task on convexity and dual spaces

    Homework Statement Let C be a non-empty convex subset of a real normed space (X,\|\cdot\|). Denote H(f,a):=\{x\in X: f(x)\leq a\} for f\in X^* (dual space) and a\in\mathbb{R}. Show that the closure \bar{C} of C satisfies \bar{C}=\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)...
  14. A

    Functional analysis - question about separable dual spaces

    Suppose X is a normed space and X*, the space of all continuous linear functionals on X, is separable. My professor claims in our lecture notes that we KNOW that X* contains functionals of arbitrarily large norm. Can someone explain how we know this, please?
  15. P

    Confused about the concepts of dual spaces, dual bases, reflexivity and annihilators

    My background in linear algebra is pretty basic: high school math and a first year course about matrix math. Now I'm reading a book about finite-dimensional vector spaces and there are a few concepts that are just absolutely bewildering to me: dual spaces, dual bases, reflexivity and...
  16. S

    Linear Functionals, Dual Spaces & Linear Transformations Between Them

    I have a question about mappings that go from a vector space to the dual space, the notation is quite strange. A linear functional is just a linear map f : V → F. The dual space of V is the vector space L(V,F) = (V)*, i.e. the space of linear functionals, i.e. maps from V to F. L(V,F)=...
  17. W

    Natural isomorphisms between dual spaces

    EDIT: finite dimensional only! Hello, I would like to ask a question; I understand that the cannonical "evaluation map" ( (p(v))(f) = f(v) , f is a functional, v in V ) from V -> V** is a "natural" isomorphism ( we don't have to select any bases, the isomorphism relies on no choices ), so V...
  18. P

    What is the concept of Dual Spaces and how do they relate to vector spaces?

    So I asked this question about Rigged Hilbert Space https://www.physicsforums.com/showthread.php?t=435123 And one of the problem I have understand Rigged Hilbert Space is that it involves taking the dual of a particular dense subspace of Hilbert Space and I of course have no clue what the...
  19. S

    Linear Algebra - Dual Spaces

    [SOLVED] Linear Algebra - Dual Spaces Homework Statement (V and W are vector spaces. F is a field) "The space L(V,W) of linear maps from V to W is always a vector space. Take W = F. We then get the space V* := L(V,F) of F-linear maps V --> F. This is called the dual space of V." 1. Let V =...
  20. D

    Dual Spaces of Vector Spaces: Conventions and Dual Bases

    Whilst trying to refresh myself on what a dual space of a vector space is I have confused myself slightly regarding conventions. (I am only bothered about finite dimensional vector spaces.) I know what a vector space, a dual space and a basis of a vector space are but dual bases: I seem...