= Dual space
{title2=$V^*=\operatorname{Hom}_F(V,F)$}
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= Dual vector space
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= Algebraic dual
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The <dual space> of a <vector space> $V$ over a <field> $F$ is the <vector space> of all <linear functionals> $\varphi:V\to F$. Think of a <vector> as an input and a <linear functional> as a scalar-valued measurement of that input. For $V=\mathbb R^2$, the rules $(x,y)\mapsto x$, $(x,y)\mapsto y$, and $(x,y)\mapsto2x-y$ are different elements of the <dual space>. None needs a notion of <length> or <angle>.
Its operations combine measurements pointwise:
$$
(\varphi+\psi)(v)=\varphi(v)+\psi(v),\qquad
(a\varphi)(v)=a\varphi(v).
$$
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 $n$-dimensional <vector space> has an $n$-dimensional <dual space>. The spaces have the same <dimension>, but their elements play different roles: an input and a rule that measures an input. <Linear Algebra Done Right by Sheldon Axler (4th edition)>[Axler, Linear Algebra Done Right, fourth edition, §3F, definitions 3.109–3.112]{ref}.
This separation is useful when changing coordinates. The <dual pairing> $\varphi(v)$ must stay the same while the component lists for $v$ and $\varphi$ 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.
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