= Linear functional
{title2=$\varphi:V\to F$}
{wiki}
= Linear form
{synonym}
A <linear functional> is a <linear map> from a <vector space> $V$ to its scalar <field> $F$:
$$
\varphi(av+bw)=a\varphi(v)+b\varphi(w),\qquad a,b\in F,\quad v,w\in V.
$$
Its output is one <scalar>, rather than an arbitrary <vector> in another space. In $\mathbb R^2$, $\varphi(x,y)=2x-y$ gives $\varphi(3,1)=5$, $\varphi(1,2)=0$, and $\varphi(4,3)=5$: adding the two inputs adds their measured values.
The <level sets> $\varphi(v)=c$ visualize what the measurement retains and discards. For this example they are parallel <straight lines> $y=2x-c$. Moving parallel to $(1,2)$ changes the input but leaves $\varphi$ unchanged. The zero <level set> is its <kernel of a linear map>, while the other <level sets> are translates of that <vector subspace>.
\Image[/linear-functional-level-sets.png]
{title=A <linear functional> groups <vectors> into parallel <level sets> with the same measured value}
Inputs need not be geometric arrows. On the <vector space> of <polynomials> of <polynomial degree> at most two, evaluating at zero is a <linear functional>, as is taking the <definite integral> from zero to one. For $p(t)=a+bt+ct^2$, they give $p(0)=a$ and $\int_0^1p(t)\,dt=a+b/2+c/3$. Thus <polynomial evaluation> and <integration> are two different measurements of the same <polynomial>. Coordinate extraction is another measurement, developed under <dual basis>.
A <linear functional> is defined by <linearity>, not by <continuity>. In finite-dimensional real or complex <normed vector spaces> it is automatically continuous; in infinite <dimension> it need not be. The <continuous dual space> adds this requirement when small changes in the input should give small changes in the output.
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