A recurring theme in mathematics is that of duality: a mathematical object {X}can either be described internally (or in physical space, or locally), by describing what {X} physically consists of (or what kind of maps exist into {X}), or externally(or in frequency space, or globally), by describing what {X} globally interacts or resonates with (or what kind of maps exist out of {X}). These two fundamentally opposed perspectives on the object {X} are often dual to each other in various ways: performing an operation on {X} may transform it one way in physical space, but in a dual way in frequency space, with the frequency space description often being a "inversion" of the physical space description. In several important cases, one is fortunate enough to have some sort offundamental theorem connecting the internal and external perspectives. Here are some (closely inter-related) examples of this perspective:

  1. Vector space duality A vector space {V} over a field {F} can be described either by the set of vectors inside {V}, or dually by the set of linear functionals {\lambda: V \rightarrow F} from {V} to the field {F} (or equivalently, the set of vectors inside the dual space {V^*}). (If one is working in the category of topological vector spaces, one would work instead with continuous linear functionals; and so forth.) A fundamental connection between the two is given by the Hahn-Banach theorem (and its relatives).
  2. Vector subspace duality In a similar spirit, a subspace {W} of {V} can be described either by listing a basis or a spanning set, or dually by a list of linear functionals that cut out that subspace (i.e. a spanning set for the orthogonal complement {W^\perp := \{ \lambda \in V^*: \lambda(w)=0 \hbox{ for all } w \in W \})}. Again, the Hahn-Banach theorem provides a fundamental connection between the two perspectives.
  3. Convex duality More generally, a (closed, bounded) convex body {K} in a vector space {V} can be described either by listing a set of (extreme) points whose convex hull is {K}, or else by listing a set of (irreducible) linear inequalities that cut out {K}. The fundamental connection between the two is given by the Farkas lemma.
  4. Ideal-variety duality In a slightly different direction, an algebraic variety {V} in an affine space {A^n} can be viewed either "in physical space" or "internally" as a collection of points in {V}, or else "in frequency space" or "externally" as a collection of polynomials on {A^n} whose simultaneous zero locus cuts out {V}. The fundamental connection between the two perspectives is given by the nullstellensatz, which then leads to many of the basic fundamental theorems in classical algebraic geometry.
  5. Hilbert space duality An element {v} in a Hilbert space {H} can either be thought of in physical space as a vector in that space, or in momentum space as a covector {w \mapsto \langle v, w \rangle} on that space. The fundamental connection between the two is given by the Riesz representation theorem for Hilbert spaces.
  6. Semantic-syntactic duality Much more generally still, a mathematical theory can either be described internally or syntactically via its axioms and theorems, or externally or semantically via its models. The fundamental connection between the two perspectives is given by the Gödel completeness theorem.
  7. Intrinsic-extrinsic duality A (Riemannian) manifold {M} can either be viewed intrinsically (using only concepts that do not require an ambient space, such as the Levi-Civita connection), or extrinsically, for instance as the level set of some defining function in an ambient space. Some important connections between the two perspectives includes the Nash embedding theorem and the theorema egregium.
  8. Group duality A group {G} can be described either via presentations (lists of generators, together with relations between them) or representations(realisations of that group in some more concrete group of transformations). A fundamental connection between the two is Cayley's theorem. Unfortunately, in general it is difficult to build upon this connection (except in special cases, such as the abelian case), and one cannot always pass effortlessly from one perspective to the other.
  9. Pontryagin group duality A (locally compact Hausdorff) abelian group {G}can be described either by listing its elements {g \in G}, or by listing thecharacters {\chi: G \rightarrow {\bf R}/{\bf Z}} (i.e. continuous homomorphisms from {G} to the unit circle, or equivalently elements of {\hat G}). The connection between the two is the focus of abstract harmonic analysis.
  10. Pontryagin subgroup duality A subgroup {H} of a locally compact abelian group {G} can be described either by generators in {H}, or generators in the orthogonal complement {H^\perp := \{ \xi \in \hat G: \xi \cdot h = 0 \hbox{ for all } h \in H \}}. One of the fundamental connections between the two is the Poisson summation formula.
  11. Fourier duality A (sufficiently nice) function {f: G \rightarrow {\bf C}} on a locally compact abelian group {G} (equipped with a Haar measure {\mu}) can either be described in physical space (by its values {f(x)} at each element {x} of {G}) or in frequency space (by the values {\hat f(\xi) = \int_G f(x) e( - \xi \cdot x )\ d\mu(x)}at elements {\xi} of the Pontryagin dual {\hat G}). The fundamental connection between the two is the Fourier inversion formula.
  12. The uncertainty principle The behaviour of a function {f} at physical scales above (resp. below) a certain scale {R} is almost completely controlled by the behaviour of its Fourier transform {\hat f} at frequency scales below (resp. above) the dual scale {1/R} and vice versa, thanks to various mathematical manifestations of the uncertainty principle. (The Poisson summation formula can also be viewed as a variant of this principle, using subgroups instead of scales.)
  13. Stone/Gelfand duality A (locally compact Hausdorff) topological space {X}can be viewed in physical space (as a collection of points), or dually, via the {C^*} algebra {C(X)} of continuous complex-valued functions on that space, or (in the case when {X} is compact and totally disconnected) via the boolean algebra of clopen sets (or equivalently, the idempotents of {C(X)}). The fundamental connection between the two is given by theStone representation theorem or the (commutative) Gelfand-Naimark theorem.