# Tagged Questions

For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.

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### Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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### Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
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### Geometric interpretation of linear programming dual

Is there a geometric interpretation of the linear programming dual in terms of the primal? I feel like without some sort of intuition of it, I don't truly understand it.
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### Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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### Finding shortest vertical segment connecting two sets of intersecting half-planes

Consider two sets of $n$ half-planes each. Denote the sets by $A$ and $B$. How can we find a vertical segment $s$ of a minimum length such that the upper end of $s$ is in the intersection of $A$ and ...
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### Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
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### Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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### Linear independence of functionals imply null coeficients on sum?

Let $V$ be a vector space over a field $K$, $V^*$ be it's dual (it's linear functionals), $\{\alpha_1,...,\alpha_n\}$ be a basis for $V$ and $\{f_1,...,f_n\}$ be the dual basis. Any subset $S^*$ of ...
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### Relation between the dual space, transpose matrices and rank-nullity theorem

Summing up, how can one use linear functionals, transpose matrices, row and column rank equality and annihilators to prove the rank-nullity theorem? While studying linear algebra, I'm trying to get ...
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### Dual statements involving functors

I know how to construct the dual of a statement concerning objects and morphisms of a category, and understand the duality principle associated, but I am having trouble when various categories and ...
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### How can I prove algebraic and topological dual spaces do not coincide in infinite dimensional normed vector spaces?

I've heard it's enough to give an example of a non continuous linear functional, but I'm kinda confused, because some definitions ask for "bounded" at infinite spaces, does bounded mean continuous in ...
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### To prove an algebraic dual space is diferent from a topologial dual space?

I've got a normed space of infinite dimension, I've proved that if the space is finite dimensional, then the topological and algebraic dual spaces coincide, but the other way is harder, I can't do ...
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### Diference between dual spaces

What is the diference between Algebraic Dual Space and Topologic Dual Space in Normed Vector spaces with $dim=\infty$
I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...