Useful limit theorem

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22.03.2012

Here is a useful limit theorem which can be used to prove many of the commonly needed limit theorems as special cases.

Claim

Let , , , and be topological spaces, where is a subspace of . Let be such that , where and . Let be continuous at . Then

Example 1 : The limit of evaluations is the evaluation of the limit

Let be such that , and be continuous at . Then

Actually, this is equivalent to continuity at .

Example 2: The limit of sequential evaluations is the evaluation of the limit

Let , where is a subspace of . Let be a continuous function. Then

Example 3 : The limit of a component is the component of a limit

Let be such that and be such that . Since the projection is continuous,

Actually, the reverse implication also holds: if the limits of all component functions exist, then so does the limit of the function (with the relation above).

Example 4 : Semigroup addition

Let and be topological semigroups (e.g vector spaces), , and be such that . Since addition is continuous,

We also used the result of example 3 here.

Example 5 : Module multiplication

Let , , and be topological -modules (e.g. vector spaces), , and be such that , where . Since by definition the multiplication is continuous, so is its restriction to a fixed , and thus

Proof

Denote by the neighborhoods of a point (and similarly for other spaces). Let , and . Since is continuous at , there exists such that . By the definition of a limit of a function, there exists such that . Therefore , i.e. . QED.