The fundamental connection between the two perspectives is given by the Gödel completeness theorem.
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 is given by the Riesz representation theorem for Hilbert spaces. Hilbert space duality An element in a Hilbert space can either be thought of in physical space as a vector in that space, or in momentum space as a covector on that space. 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. Ideal-variety duality In a slightly different direction, an algebraic variety in an affine space can be viewed either “in physical space” or “internally” as a collection of points in, or else “in frequency space” or “externally” as a collection of polynomials on whose simultaneous zero locus cuts out. The fundamental connection between the two is given by the Farkas lemma. Convex duality More generally, a (closed, bounded) convex body in a vector space can be described either by listing a set of (extreme) points whose convex hull is, or else by listing a set of (irreducible) linear inequalities that cut out. 
Again, the Hahn-Banach theorem provides a fundamental connection between the two perspectives.
a spanning set for the orthogonal complement.
Vector subspace duality In a similar spirit, a subspace of 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. (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). 
Vector space duality A vector space over a field can be described either by the set of vectors inside, or dually by the set of linear functionals from to the field (or equivalently, the set of vectors inside the dual space ).
Here are some (closely inter-related) examples of this perspective: In several important cases, one is fortunate enough to have some sort of fundamental theorem connecting the internal and external perspectives. These two fundamentally opposed perspectives on the object are often dual to each other in various ways: performing an operation on 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. A recurring theme in mathematics is that of duality: a mathematical object can either be described internally (or in physical space, or locally), by describing what physically consists of (or what kind of maps exist into ), or externally (or in frequency space, or globally), by describing what globally interacts or resonates with (or what kind of maps exist out of ).
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