03 Supremum, Infimum, Cross Product, Complex Numbers, Sequences

Lecture from: 26.02.2024 | Video: Video ETHZ

Characterization of Supremum and Infimum

The following theorem provides a useful way to characterize the supremum of a set:

Theorem: Let $B\subseteq \mathbb{R}$ be a non-empty set bounded above. Then $r=\sup (B)$ if and only if:

1. Upper Bound: $\forall x\in B:x\le r$ ($r$ is an upper bound for $B$).
2. Approachability: $\forall ϵ>0,\exists x\in B:x>r-ϵ$ ($r$ can be approached arbitrarily closely from within $B$).

Justification:

• Necessity: If $r=\sup (B)$, then (1) must hold by definition of a supremum (least upper bound). If (2) were false, there would be some $ϵ>0$ such that no element of $B$ is greater than $r-ϵ$. This would imply that $r-ϵ$ is an upper bound of $B$, smaller than $r$, contradicting the minimality of $r$.

• Sufficiency: If (1) and (2) hold, $r$ is an upper bound. If there were a smaller upper bound $r^{\prime }<r$, choose $ϵ=r-r^{\prime }>0$. By (2), there exists $x\in B$ with $x>r-ϵ=r-(r-r^{\prime })=r^{\prime }$, contradicting that $r^{\prime }$ is an upper bound. Thus, $r$ must be the least upper bound.

A similar characterization holds for the infimum: $r=\inf (B)$ if and only if (1) $\forall x\in B:x\ge r$, and (2) $\forall ϵ>0,\exists x\in B:x<r+ϵ$.

The Cross Product: Geometry and Algebra in ${\mathbb{R}}^3$

The cross product is a unique and powerful operation specific to three-dimensional space (${\mathbb{R}}^3$). It beautifully connects geometric concepts like area and volume to algebraic computations. Unlike the dot product, which produces a scalar, the cross product of two vectors results in a new vector with special geometric properties.

Geometric Intuition: Areas and Normal Vectors

Consider two vectors $\mathbf{u}$ and $\mathbf{v}$ in ${\mathbb{R}}^3$. They define a parallelogram. The cross product, denoted $\mathbf{u}\times \mathbf{v}$, is a vector with the following key properties:

1. Direction: $\mathbf{u}\times \mathbf{v}$ is perpendicular (orthogonal) to the plane containing $\mathbf{u}$ and $\mathbf{v}$. Its direction is determined by the right-hand rule: if you curl the fingers of your right hand from $\mathbf{u}$ towards $\mathbf{v}$, your thumb points in the direction of $\mathbf{u}\times \mathbf{v}$.

2. Magnitude: The length of $\mathbf{u}\times \mathbf{v}$, denoted $||\mathbf{u}\times \mathbf{v}||$, is equal to the area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$.

Tldr

The cross product gives us a way to find a vector normal (perpendicular) to a plane defined by two vectors.

Connecting Geometry to Algebra: The Determinant Formula

How can we capture these geometric properties algebraically? The answer lies in the determinant. Let $\mathbf{u}=(u_1,u_2,u_3)$ and $\mathbf{v}=(v_1,v_2,v_3)$. The cross product $\mathbf{w}=\mathbf{u}\times \mathbf{v}$ can be computed as:

where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the standard basis vectors in ${\mathbb{R}}^3$.

Technically not the determinant

While this looks like a determinant of a $3\times 3$ matrix, it’s technically not, as the first row contains vectors, not scalars. Think of it as a convenient mnemonic for remembering the formula. The key takeaway is that the components of the cross product are combinations of the components of $\mathbf{u}$ and $\mathbf{v}$, designed to ensure orthogonality and represent the area.

Algebraic Properties of the Cross Product

The cross product has several important algebraic properties:

1. Anti-commutative: $\mathbf{u}\times \mathbf{v}=-(\mathbf{v}\times \mathbf{u})$. The order matters! Swapping the order reverses the direction of the resulting vector.

2. Distributive: $\mathbf{u}\times (\mathbf{v}+\mathbf{w})=(\mathbf{u}\times \mathbf{v})+(\mathbf{u}\times \mathbf{w})$. The cross product distributes over vector addition.

3. Scalar Multiplication: $(c\mathbf{u})\times \mathbf{v}=c(\mathbf{u}\times \mathbf{v})$. Scaling a vector scales the area of the parallelogram.

4. Not Associative: The cross product is not associative: it’s generally not true that $(\mathbf{u}\times \mathbf{v})\times \mathbf{w}=\mathbf{u}\times (\mathbf{v}\times \mathbf{w})$.

5. Jacobi Identity: $\mathbf{u}\times (\mathbf{v}\times \mathbf{w})+\mathbf{v}\times (\mathbf{w}\times \mathbf{u})+\mathbf{w}\times (\mathbf{u}\times \mathbf{v})=0$. This less intuitive identity reflects the non-associativity of the cross product.

The Cross Product and Triple Products

The cross product naturally leads to the scalar triple product, a useful tool for calculating volumes:

$\mathbf{u}\cdot (\mathbf{v}\times \mathbf{w})$

This represents the volume of the parallelepiped formed by the three vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$. Notice that we can compute this by taking the determinant of the matrix formed by placing each vector in a row (or column). The absolute value of the determinant gives the volume of the parallelepiped.

Complex Numbers: Extending the Real Number Line to a Plane

Motivation: Breaking the Barrier of $x^2+1=0$

The real numbers, while powerful, lack solutions to seemingly simple equations like $x^2+1=0$. Complex numbers elegantly address this limitation by introducing a new “imaginary” unit, denoted $i$, with the property that $i^2=-1$. This seemingly small addition unlocks a rich and powerful extension of the real numbers.

Defining Complex Numbers: Ordered Pairs with Special Multiplication

We define the set of complex numbers, denoted $\mathbb{C}$, not as a simple extension of the real line, but as a two-dimensional plane:

$\mathbb{C}=\{(x,y)\mid x,y\in \mathbb{R}\}$

Each complex number is an ordered pair of real numbers, representing a point in the complex plane. Addition is component-wise, just like with vectors in ${\mathbb{R}}^2$:

• Addition: $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$

The magic happens with multiplication:

• Multiplication: $(x_1,y_1)\cdot (x_2,y_2)=(x_1{x}_2-y_1{y}_2,x_1{y}_2+x_2{y}_1)$

This definition, while perhaps initially unintuitive, is precisely what allows us to solve $x^2+1=0$.

The Imaginary Unit and the Standard Form

Let’s consider the complex number $(0,1)$. Using the multiplication rule:

$(0,1)\cdot (0,1)=(0\cdot 0-1\cdot 1,0\cdot 1+1\cdot 0)=(-1,0)$

We identify $(-1,0)$ with the real number $-1$. Define $i=(0,1)$. Then $i^2=-1$! This is the imaginary unit. We can now express any complex number $(x,y)$ as:

$z=x+yi,\text{ where }x,y\in \mathbb{R}$

This is the standard form of a complex number, where $x$ is the real part ($\text{Re}(z)$) and $y$ is the imaginary part ($\text{Im}(z)$). We naturally identify any real number $x$ with the complex number $x+0i$.

Complex Conjugate: Reflection and a Key Operation

The complex conjugate of $z=x+yi$ is defined as $\bar{z}=x-yi$.

Why?

Geometrically, conjugation is a reflection across the real axis in the complex plane. It’s a crucial operation in complex arithmetic, allowing us to move between complex and real numbers seamlessly.

Norm (Modulus): Measuring Distance in the Complex Plane

The norm (or modulus) of a complex number $z=x+yi$ is:

$|z|=\sqrt{z\bar{z}}=\sqrt{x^2+y^2}$

This is the distance from the origin to the point $(x,y)$ in the complex plane, analogous to the magnitude of a vector in ${\mathbb{R}}^2$.

Representations: Cartesian and Polar Forms

Cartesian Form: For Addition and Subtraction

The standard form $z=x+yi$ is also known as the Cartesian form. It’s convenient for addition and subtraction because these operations are performed component-wise.

Polar Form: For Multiplication and Powers

The polar form represents a complex number in terms of its distance from the origin ($r=|z|$) and angle ($\theta$) with the positive real axis:

$z=r(\cos (\theta )+i\sin (\theta ))$

Using Euler’s formula, $e^{i\theta }=\cos (\theta )+i\sin (\theta )$, we can write the polar form more compactly as $z=r{e}^{i\theta }$.

The polar form simplifies multiplication and taking powers: if $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$, then $z_1{z}_2=(r_1{r}_2)e^{i({\theta }_1+{\theta }_2)}$.

Roots of Complex Numbers: Symmetry in the Plane

Theorem: Every non-zero complex number has exactly $n$ distinct $n$-th roots.

These roots are evenly spaced around a circle of radius $|z{|}^{1/n}$ centered at the origin in the complex plane. The formula for the $n$-th roots of $z=r{e}^{i\theta }$ is:

$z_k=r^{1/n}{e}^{i(\frac{\theta +2k\pi }{n})}$

for $k=0,1,\dots ,n-1$.

The Fundamental Theorem of Algebra: Completeness of the Complex Field

Theorem: Every non-constant polynomial with complex coefficients has at least one complex root.

This profound result highlights the “algebraic completeness” of the complex numbers. It implies that any polynomial of degree $n$ can be factored into exactly $n$ linear terms (corresponding to its $n$ roots), a property not shared by the real numbers.

Sequences and Series: Foundations

Sequences and their Limits

Definition of a Sequence

A sequence of real numbers is a function $a:{\mathbb{N}}^{\ast }\to \mathbb{R}$, where ${\mathbb{N}}^{\ast }=\{1,2,3,\dots \}$. We denote the sequence as $(a_n{)}_{n\ge 1}$, where $a_n=a(n)$.

Examples of Sequences

1. $a_n=\frac{1}{n}$. This sequence converges to 0.
2. $a_n=(-1{)}^n$. This sequence oscillates between -1 and 1 and does not converge.

Tldr

You can visualize a sequence by plotting its terms on a graph. For a converging sequence, the points will cluster around the limit as n increases.

Uniqueness of the Limit (Lemma)

Let $(a_n{)}_{n\ge 1}$ be a sequence. There is at most one $l\in \mathbb{R}$ such that for all $ϵ>0$, the set $\{n\in {\mathbb{N}}^{\ast }\mid a_n\notin (l-ϵ,l+ϵ)\}$ is finite. This means that outside any small interval around $l$, there are only finitely many terms of the sequence.

Proof: Suppose there are two distinct limits, $l_1$ and $l_2$. Choose $ϵ=\frac{|l_1-l_2|}{2}>0$. Then the intervals $(l_1-ϵ,l_1+ϵ)$ and $(l_2-ϵ,l_2+ϵ)$ are disjoint. By the definition of a limit, there would only be finitely many terms outside each interval. But this means there would only be finitely many terms in total, contradicting the fact that the sequence is infinite.

Convergence of a Sequence

Definition: Capturing the Essence of Approaching a Limit

A sequence $(a_n{)}_{n\ge 1}$ of real numbers converges to a limit $l\in \mathbb{R}$ if for any arbitrarily small positive distance $ϵ$, we can find a point in the sequence beyond which all subsequent terms are within that distance of the limit. More formally:

Theorem

A sequence $(a_n{)}_{n\ge 1}$ converges to a limit $l\in \mathbb{R}$ if for all $ϵ>0$, there exists an index $N\in \mathbb{N}$ such that for all $n>N$ (not $n\ge N$) , the distance between $a_n$ and $l$ is less than $ϵ$:

$|a_n-l|<ϵ$

This definition captures the intuitive idea of a sequence “approaching” a limit: as we go further down the sequence (i.e., as $n$ increases), the terms get arbitrarily close to the limit $l$.

Uniqueness of the Limit

Lemma: The limit of a convergent sequence is unique.

Proof: (By contradiction) Suppose a sequence $(a_n)$ converges to two distinct limits, $l_1$ and $l_2$. Let $d=|l_1-l_2|>0$ be the distance between these limits. Choose $ϵ=\frac{d}{2}$. By the definition of convergence, there exist $N_1$ and $N_2$ such that:

• For $n>N_1$, $|a_n-l_1|<ϵ$
• For $n>N_2$, $|a_n-l_2|<ϵ$

Let $N=\max (N_1,N_2)$. Then, for $n>N$, both inequalities hold. By the triangle inequality:

$|l_1-l_2|=|(l_1-a_n)+(a_n-l_2)|\le |l_1-a_n|+|a_n-l_2|<ϵ+ϵ=d$

But this contradicts our assumption that $|l_1-l_2|=d$. Therefore, the limit must be unique.

If a limit exists, we denote it by: ${\lim }_{n\to \infty }{a}_n=l$

Boundedness: Convergence Implies Boundedness

Remark: Every convergent sequence is bounded.

Intuition

If a sequence converges to a limit, its terms eventually cluster around that limit. This clustering implies that the terms cannot grow arbitrarily large or small.

Proof: Let $(a_n)$ converge to $l$. Choose $ϵ=1$. There exists $N$ such that for $n>N$, $|a_n-l|<1$. This implies $|a_n|<|l|+1$. Let $M=\max \{|a_1|,|a_2|,\dots ,|a_N|,|l|+1\}$. Then $|a_n|\le M$ for all $n$, so the sequence is bounded.

An Equivalent Definition: Easier to Use in Proofs

Lemma: The following statements are equivalent:

1. $(a_n{)}_{n\ge 1}$ converges to $l$.
2. For all $ϵ>0$, there exists $N\in \mathbb{N}$ such that for all $n>N$, $|a_n-l|<ϵ$.

Why?

While subtly different from the original definition, this version is often easier to use in proofs. It allows us to focus on finding a suitable $N$ for a given $ϵ$ without explicitly dealing with the set of indices.

Example: Proving Convergence with the Archimedean Principle

Let $a_n=\frac{n}{n+1}$. We want to prove that ${\lim }_{n\to \infty }{a}_n=1$.

Proof: We need to show that for any $ϵ>0$, there exists an $N$ such that for all $n>N$, $|a_n-1|<ϵ$.

$|a_n-1|=\left|\frac{n}{n+1}-1\right|=\left|\frac{-1}{n+1}\right|=\frac{1}{n+1}$

Given $ϵ>0$, by the Archimedean Principle, there exists an integer $N$ such that $N>\frac{1}{ϵ}-1$. Then, for all $n>N$, we have $n+1>N+1>\frac{1}{ϵ}$, so $\frac{1}{n+1}<ϵ$. Thus, $|a_n-1|<ϵ$ for all $n>N$. This proves that ${\lim }_{n\to \infty }{a}_n=1$.

Continue here: 04 Divergence, Limits, Monotony, Weierstrass Theorem, Limit Superior and Inferior