Quartz sync: May 26, 2025, 2:39 PM

content/Definitions/Metric Spaces/Ball.md

@@ -1,6 +1,6 @@
 # Definition
 Let $d$ be a [[Metric|metric]] on a set $X$.
 
-The (open) **ball** with centre $x \in X$ and radius $r \geq 0$ is $B_{r} \equiv \{ y \in X | d(x,y) \gt r\}$.
+The (open) **ball** with centre $x \in X$ and radius $r \gt 0$ is $B_{r} \equiv \{ y \in X | d(x,y) \gt r\}$.
 
 A sequence $\{ X_{n} \}$ in $X$ **converges** to $x \in X$ if it eventually belongs to any ball $B_{r}(x)$; $\forall r \gt 0 \; \exists N \in \mathbb{N}$ such that $\underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N$.

content/Definitions/Metric Spaces/Complete.md

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+# Definition
+A [[Metric Space|metric space]] is called **complete** (or a **Cauchy space**) if every [[Cauchy Sequence|Cauchy sequence]] of points in $M$ has a limit that is also in $M$.
+
+# Example
+See [[Cauchy Sequence#Cauchy Sequence]]

content/Definitions/Norm.md

@@ -1,5 +1,5 @@
 # Definition
-A **norm** on $V$ is a map $\|\cdot\| : V \to [ \, 0, \infty \rangle$ such that
+A **norm** on $V$ is a map $\|{\cdot}\| : V \to [ \, 0, \infty \rangle$ such that
 1. $\| u + v \| \leq \| u \| + \| v \|, \; \forall u,v \in V$
 2. $\| c \times v \| = | c | \times \| v \|, \; \forall v \in V, \, \forall c \in \mathbb{C}$
 3. $\| v \| = 0 \implies v = 0$

content/Definitions/Vector Spaces/Banach Space.md

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+# Definition
+A **Banach space** is a [[complete]] [[Normed Vector Space|normed vector space]] $(X, \|{\cdot}\|)$.
+

content/Exam Preparation.md

@@ -9,9 +9,9 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 2. What is an equivalence relation? [[Question 2|Answer]]
 3. What is a topological space? Examples? [[Question 3|Answer]]
 4. What is the ball topology on a metric space? [[Question 4|Answer]]
-5. What is the topology on a Banach space?
-6. What is a compact set?
-7. State the Heine-Borel theorem. Proof?
+5. What is the topology on a Banach space? [[Question 5|Answer]] - TODO
+6. What is a compact set? [[Question 6|Answer]]
+7. State the Heine-Borel theorem. Proof? [[Question 7|Answer]]
 8. What is a continuous function?
 9. Why does a real valued continuous function obtain its maximum on a compact set?
 10. What is a net? Given an example of an upward filtered ordered set.

content/Lectures/Lecture 3.md

@@ -6,7 +6,7 @@ date: 2025-01-13
 Each real number is a limit of a sequence of rational numbers.
 $$\mathbb{Q} \subset \mathbb{R}$$
 As an ordered [[Number Field]] $\mathbb{R}$ is **complete**, meaning that every [[Cauchy Sequence]] in $\mathbb{R}$ converges to a real number.
-Equivalently, the real numbers have the **[[Least Upper Bound Property]]**; $\forall X \subset \mathbb{R}$ bounded above has a least upper bound denoted by $sup(X) \in \mathbb{R}$. Eq. an $inf(Y) \in \mathbb{R}$ if $Y$ bounded below.
+Equivalently, the real numbers have the **[[Least Upper Bound Property]]**; $\forall X \subset \mathbb{R}$ bounded above has a least upper bound denoted by $sup(X) \in \mathbb{R}$. Eq. an $\inf(Y) \in \mathbb{R}$ if $Y$ bounded below.
 # Functions and Cardinality
 A function $f : X \to Y$ is **[[Injective]]** if $f(x) = f(y) \implies x = y$.
 

content/Revision/Real Analysis/Question 4.md

@@ -1,3 +1,24 @@
 # Question
 What is the ball topology on a metric space?
 # Answer
+A [[Ball|ball]] has two types, either an open (denoted by $B_{r}(x)$) or closed ball (denoted by $B_{r}[x]$).
+
+Let $d$ be a [[Metric|metric]], on a set $X$, giving us a [[Metric Space|metric space]] $(X,d)$.
+
+And a ball has two properties:
+- A radius $r \gt 0$
+- Centre point $x \in X$.
+Then the ball is defined as a set of points in $X$ that is of distance less than $r$ away from $x$
+$$B_{r}(x) = \{ y \in X\ |\ d(y,x) \lt r\}.$$
+## Closed Ball
+> [!note]
+> I assume this is not required as I don't think this was covered in the lecture.
+
+A closed [[Ball|ball]] is similar to an open ball. However includes the points on the boundary.
+
+It has similar properties as the open ball, but it is defined as the points less than or equal to $r$ away from $p$
+$$B_{r}[x] = \{ y \in X\ |\ d(y, x) \leq r \}.$$
+> [!info] In a nutshell
+> Replaces the closed one replaces the $\lt$ with $\leq$ in $d(x,y) \lt r$.
+
+

content/Revision/Real Analysis/Question 5.md

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+# Question
+What is the topology on a Banach space?
+# Answer

content/Revision/Real Analysis/Question 6.md

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+# Question
+What is a compact set?
+# Answer
+$X$ is [[Compact|compact]] if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]]
+
+That means that there is an [[Open Sets|open set]] $A \subset X$ in $(X, d)$ which consists of only interior points, where a union with $X$ creates an [[Open Cover|open cover]]. 
+
+This [[Open Cover|open cover]] has any subcollection with union $X$.
+
+> [!info] What is a subcollection?
+> A subcollection is a synonym for a "set" or "subset".

content/Revision/Real Analysis/Question 7.md

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+# Question
+State the Heine-Borel theorem. Proof?
+# Answer
+## Heine-Borel
+The Heine-Borel theorem states that if there is a subset $A \subset \mathbb{R}^{n}$.
+Then $A$ is [[Compact|compact]], if and only if, $A$ is closed and [[Bounded|bounded]].
+## Proof
+### For $\Rightarrow$
+$A$ is closed since $\mathbb{R}^{n}$ is [[Hausdorff]] and $A$ is [[Compact|compact]]
+
+It is [[Bounded|bounded]] since we can cover it by finitely many [[Ball|balls]].
+### For $\Leftarrow$
+Assume first that $A$ is an $n$-cube with boundary included, and say that $A$ is not [[Compact|compact]].
+
+If an [[Open Cover|open cover]] of $A$ has no finite [[Subcover|subcover]], then by halving sides of cubes we get sequences of cubes contained in each other, each having no finite [[Subcover|subcover]].
+
+The centres of these cubes form a [[Cauchy Sequence|Cauchy sequence]] with a limit $x \in A$.
+
+![[Pasted image 20250130121836.png]]
+$$d(x_{n}, x_{m}) \lt \varepsilon, \ \forall n,m \gt N$$
+Now we have to show that $R^{n}$ is complete.
+
+Any neighbourhood of $x$ from the [[Open Cover|cover]] will contain a small enough cube.
+
+However, this will be a finite [[Subcover|subcover]], which is a contradiction.
+
+QED.