= Continuous dual space
{title2=$X^*$}
The <continuous dual space> $X^*$ of a <topological vector space> is the <vector space> of its <continuous linear functionals>. Unlike the <algebraic dual>, it requires that a small change in an input produce a small change in its scalar measurement. For a <normed vector space>, this is equivalent to a finite bound
$$
|\varphi(x)|\leq C\lVert x\rVert
$$
for all <vectors> $x$ and some constant $C\geq0$. The least such constant is the <operator norm> of $\varphi$.
This is the continuity-restricted subspace of the algebraic <dual space>; in infinite dimension the two can differ.
For the <linear functional> $\varphi(x,y)=2x-y$ on $\mathbb R^2$ with its <Euclidean norm>, the <Cauchy-Schwarz inequality> gives
$$
|\varphi(x,y)|\leq\sqrt5\sqrt{x^2+y^2}.
$$
Equality holds on the <unit vector> $(2,-1)/\sqrt5$, so its <operator norm> is exactly $\sqrt5$. The same estimate controls measurement error: $|\varphi(v+h)-\varphi(v)|\leq\sqrt5\lVert h\rVert_2$. The <linear functional> diagram shows which changes are entirely invisible to this measurement; its <kernel of a linear map> consists of the changes with zero response. All <linear functionals> on a finite-dimensional real or complex <normed vector space> are continuous, but <discontinuous functional detected by vanishing sequence tails> shows why the restriction matters in infinite <dimension>.
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