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For a function to have an inverse, it must be one-to-one.If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function = defined on the whole of is not one-to-one since = for any .
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
Restriction to a normal subgroup behaves particularly well and is often called Clifford theory after the theorem of A. H. Clifford. [1] Restriction can be generalized to other group homomorphisms and to other rings. For any group G, its subgroup H, and a linear representation ρ of G, the restriction of ρ to H, denoted
Imagine, for instance, defining a function : by picking each value () completely at random (so its graph would be appear as infinitely many points scattered randomly about the plane ); no matter how you ended up imagining it, the Blumberg theorem guarantees that even this function has some dense subset on which its restriction is continuous.
In mathematics, a corestriction [1] of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting.It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them.
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The co-restriction of to the general linear group of a -invariant subspace is known as a subrepresentation. A representation ρ : G → G L ( V ) {\displaystyle \rho :G\to GL(V)} is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial G {\displaystyle G} -invariant ...