Lecture 1: The Setting - A Review of Euclidean Space

KSM3E04: Analysis on Manifolds — Fall 2025   |   August 11, 4:00–5:30 PM

Welcome to Analysis on Manifolds. The central goal of this course is to generalize the fundamental concepts of calculus to functions on curved spaces. To do so, we must first have a firm grasp of the analytical and topological structure of our primary model space, $\mathbf{R}^n$.

This lecture serves as an overview of the concepts from real analysis and linear algebra that will be used freely throughout the course. We assume familiarity with these topics; this is not a comprehensive treatment but rather a review to establish notation and highlight the key results we will rely on. We will focus specifically on how these foundational ideas apply to the setting of $\mathbf{R}^n$.

1. The Algebraic and Metric Structure of Euclidean Space

Euclidean $n$-space, $\mathbf{R}^n$, is the set of all $n$-tuples $x = (x^1, \dots, x^n)$ of real numbers. It forms a vector space and, to facilitate analysis, is equipped with a norm and inner product.

Definition 1 (Norm and Inner Product).

For vectors $x, y \in \mathbf{R}^n$, we define:

  1. The inner product: $ \langle x, y \rangle = \sum_{i=1}^n x^i y^i. $
  2. The Euclidean norm ($\ell^2$ norm): $ \|x\| = \sqrt{\langle x, x \rangle} = \sqrt{\sum_{i=1}^n (x^i)^2}. $

The inner product is the fundamental algebraic structure that gives $\mathbf{R}^n$ its geometric properties. It defines the Euclidean norm ($\|x\| = \sqrt{\langle x, x \rangle}$), which in turn defines the distance between points ($d(x,y) = \|x-y\|$). This hierarchy means that our entire geometric framework is derived from a purely algebraic object. The study of properties that depend only on the inner product is therefore "algebraically invariant." The Euclidean norm is the most natural choice because it is the one induced by this fundamental algebraic structure.

Theorem 1 (Fundamental Properties).

For any $x, y, z \in \mathbf{R}^n$ and $a \in \mathbf{R}$:

  • Inner Product Properties:
    • Symmetry: $\langle x, y \rangle = \langle y, x \rangle$
    • Bilinearity: $\langle ax+y, z \rangle = a\langle x, z \rangle + \langle y, z \rangle$
    • Positive Definiteness: $\langle x, x \rangle \ge 0$, and $\langle x, x \rangle = 0 \iff x=0$.
  • Cauchy-Schwarz Inequality: $|\langle x, y \rangle| \le \|x\| \|y\|$.
  • Triangle Inequality: $\|x+y\| \le \|x\| + \|y\|$.
  • Polarization Identity: $\langle x,y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2)$.
  • Pythagorean Theorem: If $\langle x, y \rangle = 0$, then $\|x+y\|^2 = \|x\|^2 + \|y\|^2$.

Theorem 2 (Equivalence of Norms in $\mathbf{R}^n$).

Let $\|\cdot\|_a$ and $\|\cdot\|_b$ be any two norms on $\mathbf{R}^n$. Then there exist positive constants $c_1$ and $c_2$ such that for all $x \in \mathbf{R}^n$:

$$ c_1 \|x\|_a \le \|x\|_b \le c_2 \|x\|_a. $$
Proof

It suffices to show that any norm $\|\cdot\|$ is equivalent to the max norm, $\|x\|_\infty = \max_i |x^i|$.

Part 1: Show $\|\cdot\| \le C \|\cdot\|_\infty$. Let $x = \sum_{i=1}^n x^i e_i$. By the triangle inequality and homogeneity:

$$ \|x\| = \left\|\sum x^i e_i\right\| \le \sum \|x^i e_i\| = \sum |x^i| \|e_i\| \le \left(\max_j |x^j|\right) \left(\sum \|e_i\|\right). $$

Letting $C = \sum \|e_i\|$, we have $\|x\| \le C \|x\|_\infty$. This also shows that the function $f(x) = \|x\|$ is continuous with respect to the topology induced by $\|\cdot\|_\infty$.

Part 2: Show $c \|\cdot\|_\infty \le \|\cdot\|$. Let $S = \{x \in \mathbf{R}^n \mid \|x\|_\infty = 1\}$ be the unit sphere in the max norm. $S$ is closed and bounded, and thus compact by the Heine-Borel theorem. The function $f(x) = \|x\|$ is continuous on $S$ and so, by the Extreme Value Theorem, it must attain a minimum value $c$ on $S$. Since $0 \notin S$, we must have $c > 0$.

For any non-zero vector $y \in \mathbf{R}^n$, the vector $y/\|y\|_\infty$ lies on $S$. Therefore:

$$ \left\| \frac{y}{\|y\|_\infty} \right\| \ge c \implies \frac{1}{\|y\|_\infty} \|y\| \ge c \implies \|y\| \ge c \|y\|_\infty. $$

Combining both parts gives $c \|x\|_\infty \le \|x\| \le C \|x\|_\infty$, proving equivalence.

2. Basic Topology of $\mathbf{R}^n$

The norm induces a metric, which defines the standard topology on $\mathbf{R}^n$. This allows us to generalize concepts like limits, continuity, and compactness from $\mathbf{R}$ to $\mathbf{R}^n$.

Definition 2 (Topological Concepts).

  1. The distance between $x,y \in \mathbf{R}^n$ is $d(x,y) = \|x-y\|$.
  2. An open ball is a set $B(a,r) = \{x \in \mathbf{R}^n \mid \|x-a\| < r\}$.
  3. A set $U \subseteq \mathbf{R}^n$ is open if for every $a \in U$, there exists $r>0$ such that $B(a,r) \subseteq U$.
  4. A set $C \subseteq \mathbf{R}^n$ is closed if its complement $\mathbf{R}^n \setminus C$ is open.
  5. A sequence $(x_k)$ in $\mathbf{R}^n$ converges to $x$ if $\lim_{k\to\infty} \|x_k - x\| = 0$.

A crucial insight is that the collection of open sets (the topology) on $\mathbf{R}^n$ does not depend on which norm we use. For example, the max norm $\|x\|_\infty = \max_i |x^i|$ defines "open balls" that are actually open hyperrectangles. The Theorem on Equivalence of Norms guarantees that for any open ball defined by the Euclidean norm, we can fit an open rectangle inside it, and vice-versa. This means that both norms generate the exact same collection of open sets. Therefore, all analytical concepts that depend on the notion of openness—such as convergence, continuity, and compactness—are independent of the specific choice of norm. The analysis depends only on the topology.

Theorem 3 (Topological Properties and Characterizations).

  • The union of any collection of open sets is open. The intersection of any finite collection of open sets is open.
  • The intersection of any collection of closed sets is closed. The union of any finite collection of closed sets is closed.
  • A set $C$ is closed if and only if for every convergent sequence $(x_k)$ with $x_k \in C$ for all $k$, the limit is also in $C$.
  • Bolzano-Weierstrass Theorem: Every bounded sequence in $\mathbf{R}^n$ has a convergent subsequence.
  • Heine-Borel Theorem: A set $K \subseteq \mathbf{R}^n$ is compact if and only if it is closed and bounded.

Definition 3 (Continuity).

Let $A \subseteq \mathbf{R}^n$ and $f: A \to \mathbf{R}^m$. We say $f$ is continuous at $a \in A$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in A$ with $\|x-a\| < \delta$, we have $\|f(x)-f(a)\| < \epsilon$.

Theorem 4 (Properties of Continuous Functions).

  • Sequential Characterization: A function $f: A \to \mathbf{R}^m$ is continuous at $a \in A$ if and only if for every sequence $(x_k)$ in $A$ that converges to $a$, the sequence $(f(x_k))$ converges to $f(a)$.
  • Topological Characterization: A function $f: A \to \mathbf{R}^m$ is continuous if and only if for every open set $V \subseteq \mathbf{R}^m$, the preimage $f^{-1}(V)$ is open in $A$.
  • Extreme Value Theorem: If $K \subseteq \mathbf{R}^n$ is compact and $f: K \to \mathbf{R}$ is continuous, then $f$ attains a maximum and minimum value on $K$.
  • The image of a compact set under a continuous map is compact.
  • The image of a connected set under a continuous map is connected.
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