Real Analysis
1 Overview
Goal: To understand the Riemann integral on a function of one real variable - the simple integration you learned in hs calculus.
1. Integrate a function f over a finite interval [a,b]
* this interval is closed aka has endpoints aka NOT infinite
* this closed interval is also naturally compact
2. function f should be uniformly continuous
A function that is closed, compact and uniformly continous has a reimann integral.
Reimann integral is defined over the reals (not the rationals) because the reals are Complete. The hilbert space and banach space are also Complete.
Main concepts: closed, continuous, compact, uniformly continuous, and completeness
2 Basics
- \(\mathbb{N}\) : Naturals
- \(\mathbb{Z}\) : Integers, Naturals + Negatives
- -3,-2,-1.5,0,0.5,1…
- One can think of integers as equivalence classes or sets
- 3/2 = {3/2,6/4,9/6,…}
- This allows to use the min function
- \(\mathbb{Q}\) : \(\frac{p}{q}\) where p,q are Integers
3 Fields
Reals is a set with multiplication and addition satisfying the below:
- \((\mathbb{R}, +, 0)\) is an abelian group
- \((\mathbb{R}/\{0\}, \cdot, 1)\) is an abelian group
- Distributive law
- \(\leq\) is total order compatible with \(+, \cdot\)
- Every cauchy sequence is a convergent sequence
3.1 Absolute value
\[ \lvert x \cdot y \rvert = \lvert x \rvert \cdot \lvert y \rvert \] \[ \lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert \]
4 Dedekind cuts
Dedekind_cuts := (A,B) :: (Partition(Q),Partition(Q))- \(A,B \subseteq \mathbb{Q},\ A,B \neq \emptyset\)
- \((A,B) = 2\ partitions\ of\ Q\)
- \(A \cup B = \mathbb{Q}\)
- \(A \cap B \neq \emptyset\)
- \(a \in A \land b \in B \Rightarrow a \lt b\)
- Every element of A is less than B
- \(A\) has no largest element
A real number is a Dedekind cut.
A real number is represented by a partition of \(\mathbb{Q}\).
5 Cauchy Sequence
\(\underset{n \rightarrow \infty}{lim}a_n=b\)
6 Cauchy Condition
7 Inner product
Inner product on a vector space V = Operation on a pair of vectors in V
discriminant proof of Cauchy-Schwarz Inequality valid for all inner products on a real vector space.
All inner products define a norm but not all norms are inner products.
8 Topology
8.1 Metric Space
Metric_Space := (M,dist) :: (Set,binary_operation)
Fields of knowledge as Metric Spaces (M,d)
- Metric Space of Proofs in Math
\((M,d) := (math,dist_{math})\) - Metric Space of Reactions in Chemistry
\((M,d) := (chem,dist_{chem})\)
Mapping metric spaces of knowledge AKA analogies
- Analogies as a mapping between Metric Spaces of Knowledge
- “Reagents is-to Reaction” as “Antecedents is-to Theorem”
- \(dist_{chem}(reagents,reaction) \approx dist_{math}(antecedents,theorem)\)
8.2 Limit aka Limit point
- Given Metric Space \((M,d)\)
- \(S \subseteq M\)
- p is a limit of S iff \(\exists p_n :: sequence\) s.t. \(p_n converges to p\)
9 brightside
9.1 3
- A sequence is bounded if exist some number that bounds the sequence
- Convergent sequences implies bounded sequences but not otherway around
- A bounded sequence can rapidly oscillate meaning it is nonconvergent
10 Riemann vs Lesbesgue integral
10.1 Riemann
- Area under the curve by summing infinitely small deltas.
Problems:
- Cannot expand to higher dimensions
- Dependent on continuity
\[lim_{n \to \infty} \int^{b}_{a} f_n(x)dx \overset{?}{=} \int^{b}_{a} lim_{n \to \infty} f_n(x)dx\]
The above equality is contingent on uniform converge of the function f, for Riemann integrals BUT
we know empirically that the equality is true for far weaker constraints.
10.2 Lesbesgue
- Instead of delta on x-axis, we choose deltas on y-axis.
- visually Horizontal slices or rectangles
- Since we choose delta on y-axis we multiply it by the preImage.
Vector Space
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Normed Space
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Inner Product Space Banach Space
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Hilbert Space
