Real Analysis
Norms
The induced norm can be thought of as the maximum gain of the operator.
Sets
Open sets have a boundary which is not included in the set. By convention, we say that the empty set is open.
The opposite of an open set is a closed set.
Closed sets have a boundary which is included in the set.
Convergence
Convergence means that we can always find a finite time such that after that time, all points in the sequence stay within a specified norm ball.
A Cauchy sequence has a looser type of convergence than a convergent sequence since it only requires all elements to in the sequence to be part of the same norm ball after some time instead of requiring the sequence to get closer and closer to a single point.
Because a complete space requires that Cauchy sequences converge, all cauchy sequences are convergent in a complete space. Two important complete spaces are
Every finite dimensional vector space
A complete normed space is also called a Banach Space.
Contractions
Informally, a contraction is a function which makes distances smaller. Suppose we look at a sequence defined by iterates of a function
The contraction mapping theorem proves that contractions have a unique fixed points, and that repeatedly applying the contraction will converge to the fixed point.
Continuity
We can make the definition of continuity more restrictive by restraining the rate of growth of the function.
This captures the idea of growing slower than linear in high dimensional space.
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