The dual space of a vector space , sometimes denoted , is the vector space of all linear forms over with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:
Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector as:
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The dual space of a vector space over a field is the vector space of all linear functionals . Think of a vector as an input and a linear functional as a scalar-valued measurement of that input. For , the rules , , and are different elements of the dual space. None needs a notion of length or angle.
Its operations combine measurements pointwise:The dual basis explains the finite-dimensional structure: specifying a linear functional's values on a basis determines its value on every vector. Thus an -dimensional vector space has an -dimensional dual space. The spaces have the same dimension, but their elements play different roles: an input and a rule that measures an input. [ref]
Linear Algebra Done Right by Sheldon Axler (4th edition). Axler, Linear Algebra Done Right, fourth edition, §3F, definitions 3.109–3.112
.This separation is useful when changing coordinates. The dual pairing must stay the same while the component lists for and change in opposite ways. Covector transformation under a change of basis makes the compensation explicit. An inner product on a finite-dimensional real vector space gives one way to represent each linear functional by a vector, but that representation uses the chosen inner product; the dual space itself does not require one.
Here dual space means the algebraic dual: all linear functionals. For a topological vector space, the continuous dual space keeps only the continuous linear functionals. In infinite dimension, this can be a proper vector subspace; discontinuous functional detected by vanishing sequence tails gives an example. The distinction matters when a scalar-valued measurement must respond continuously to small input changes.
In mathematics, particularly in functional analysis and linear algebra, the concept of the **dual space** is important in studying vector spaces and linear maps. ### Definition Given a vector space \( V \) over a field \( F \) (commonly the real numbers \( \mathbb{R} \) or complex numbers \( \mathbb{C} \)), the **dual space** \( V^* \) is defined as the set of all linear functionals on \( V \).