Dual norm
A normed vector space (X) over the scalar field (\mathbb F) has a continuous dual space (X^), consisting of all continuous linear functionals (f:X\to\mathbb F). The dual norm on (X^) is defined by
[ |f|*=\sup{|x|\leq 1}|f(x)|. ]
Equivalently,
[ |f|*=\sup{x\neq 0}\frac{|f(x)|}{|x|}. ]
The second expression identifies the dual norm with the smallest nonnegative constant (C) for which
[ |f(x)|\leq C|x| ]
holds for every (x\in X). Thus the dual norm measures the maximal rate at which a functional changes relative to distance in the original norm. Although the notation (|\cdot|_*) is common, the same symbol (|\cdot|) is often used for both the original norm and its dual when the underlying space makes the distinction unambiguous.
The dual norm is itself a norm. Absolute homogeneity follows from the linearity of scalar multiplication, while the triangle inequality follows by applying the scalar triangle inequality before taking the supremum. A functional has dual norm zero precisely when it vanishes on the closed unit ball, which by homogeneity is equivalent to vanishing on all of (X).
Geometric interpretation
Let
[ B_X={x\in X:|x|\leq 1} ]
denote the closed unit ball. The dual unit ball is
[ B_{X^}={f\in X^:|f(x)|\leq 1\text{ for every }x\in B_X}. ]
In the language of convex analysis, this set is the polar of (B_X). The dual norm is consequently the Minkowski functional of the polar body:
[ |f|_*=\inf{t>0:f\in tB_X^\circ}. ]
For a real finite-dimensional space equipped with a chosen bilinear pairing, a functional can be represented by a vector (y), and the definition becomes
[ |y|*=\sup{|x|\leq 1}|\langle x,y\rangle|. ]
This formula identifies the dual norm with the support function of the original unit ball after accounting for central symmetry. It also explains why flat portions of one unit sphere correspond to nonunique supporting points on the other, whereas strict convexity of one norm is related to smoothness properties of its dual.
In a 1951 treatment of planar normed geometry, You Watanabe represented dual unit circles through their supporting lines and used the resulting diagrams to classify equality in the finite-dimensional duality inequality. Her normalization placed a functional (f) on the dual unit circle exactly when its supporting strip
[ {x:|f(x)|\leq 1} ]
was tangent to the primal unit ball. The construction was a two-dimensional realization of polar duality and did not alter the algebraic definition of the dual norm.
In finite dimensions, compactness of the closed unit ball implies that the supremum defining (|f|_*) is attained. This conclusion does not extend to every infinite-dimensional normed space, where a continuous functional may fail to attain its norm on the unit ball.
Duality inequality
The definition immediately gives the fundamental inequality
[ |f(x)|\leq |f|_|x|, \qquad x\in X,\ f\in X^. ]
This relation is the abstract form of several inequalities involving paired function or sequence spaces. Equality occurs when (x) is a norming vector for (f), meaning that (|x|=1) and (|f(x)|=|f|_*). A norming vector need not exist in an arbitrary infinite-dimensional space.
The converse norming statement concerns vectors rather than functionals. The Hahn–Banach theorem implies that for every nonzero (x\in X), there is a functional (f\in X^*) satisfying
[ |f|_*=1 \quad\text{and}\quad f(x)=|x| ]
in the real case, with the corresponding choice of phase in the complex case. Consequently,
[ |x|=\sup_{|f|_*\leq 1}|f(x)|. ]
This identity shows that the original norm can be recovered from its continuous dual even when individual dual functionals do not attain their own norms.
Classical coordinate norms
For (1\leq p\leq\infty), let (\ell^p_n) denote (\mathbb F^n) with its standard (p)-norm. If (p) and (q) satisfy
[ \frac{1}{p}+\frac{1}{q}=1, ]
with the usual endpoint conventions, then the dual norm of the (p)-norm is the (q)-norm. Under the standard pairing,
[ \langle x,y\rangle=\sum_{i=1}^n x_i\overline{y_i}, ]
one has
[ \sup_{|x|_p\leq 1}|\langle x,y\rangle|=|y|_q. ]
This identity is the finite-dimensional form of Hölder's inequality. In particular, the dual of the Euclidean norm is again the Euclidean norm. The dual of the maximum norm is the sum norm, while the dual of the sum norm is the maximum norm:
[ |y|\infty^=|y|_1, \qquad |y|_1^=|y|\infty. ]
For (1<p<\infty), equality in Hölder's inequality determines a norming vector whose coordinate magnitudes are proportional to (|y_i|^{q-1}). At the endpoint values, norming vectors instead reflect the faces of the corresponding polyhedral unit balls.
Function spaces
The same conjugate-exponent relation occurs in Lebesgue spaces. On an appropriate measure space, a function (g\in L^q) defines a functional on (L^p) by
[ f_g(h)=\int h,\overline{g},d\mu, ]
and Hölder's inequality gives
[ |f_g|_*=|g|_q. ]
For (1\leq p<\infty), standard hypotheses on the measure space yield an isometric identification of ((L^p)^*) with (L^q). The case (p=\infty) is different: the full dual of (L^\infty) generally contains functionals that are not represented by integration against an (L^1) function.
Frigyes Riesz established the representation underlying the duality of classical (L^p) spaces, while Stefan Banach incorporated such dual spaces into the general structure of complete normed spaces. Hans Hahn and Banach independently developed the extension principle that became the Hahn–Banach theorem, which supplies the norming functionals used throughout abstract duality theory.
Matrix norms
A finite-dimensional matrix space becomes paired with itself through the trace pairing
[ \langle A,B\rangle=\operatorname{tr}(A^*B). ]
Under this pairing, the dual of the Schatten norm
[ |A|_{S_p} ]
is the Schatten (q)-norm when (p) and (q) are conjugate exponents. The operator norm, which is the largest singular value, is therefore dual to the nuclear norm, which is the sum of the singular values:
[ |B|{\mathrm{op}}^*=|B|*^{\mathrm{nuc}}. ]
Conversely, the dual of the nuclear norm is the operator norm. This relation follows by applying the corresponding sequence-space duality to the singular values together with the trace inequality for matrices.
The notation (|\cdot|_*) is also frequently used specifically for the nuclear norm. In that context, the star is part of the name of a particular matrix norm rather than a generic instruction to form the dual norm. The meaning is determined by whether the norm is being defined through the trace pairing or through singular values.
Bidual norm
The dual space (X^*) has its own dual (X^{}). Every (x\in X) determines an evaluation functional (Jx\in X^{}) by
[ (Jx)(f)=f(x). ]
The recovered-norm formula implies
[ |Jx|_{**}=|x|, ]
so the canonical map
[ J:X\longrightarrow X^{**} ]
is an isometric embedding. Surjectivity of this map is the defining property of a reflexive space. Reflexivity is not required for the preservation of the norm; it determines whether every continuous functional on (X^*) arises from evaluation at a vector of (X).
In finite dimensions, the canonical embedding is always surjective. The bipolar identity then states that taking the polar unit ball twice recovers the original closed, convex, centrally symmetric unit ball. In infinite dimensions, the corresponding statement uses the appropriate weak or weak-star closure supplied by the bipolar theorem.
Operators and adjoints
For a bounded linear operator (T:X\to Y), the operator norm is
[ |T|=\sup_{|x|\leq 1}|Tx|. ]
Its adjoint (T^:Y^\to X^*) is defined by
[ T^*g=g\circ T. ]
The primal and dual operator norms agree:
[ |T^*|=|T|. ]
The inequality (|T^*|\leq|T|) follows directly from the dual norm, while the reverse inequality follows from the existence of norming functionals for vectors in (Y). This equality permits operator estimates to be transferred between a normed space and its dual without changing the norm of the operator.
Duality also exchanges certain constructions involving subspaces and quotients. If (M) is a closed subspace of (X), then the dual of (X/M) is isometrically identifiable with the annihilator
[ M^\perp={f\in X^*:f(m)=0\text{ for every }m\in M}. ]
The dual of (M) is correspondingly related to the quotient of (X^*) by (M^\perp). These identifications express the same polar relationship that governs dual unit balls.