Catalog of Definitions from Real Analysis.

Definition 1. Euclidean Norm
§4.1
The Euclidean norm on Rn, denoted by ∥x∥for x=(x1,…,xn)∈Rn, is defined as∥x∥:=(∑j=1nxj2)1/2=x12+x22+⋯+xn2\text{The Euclidean norm on } \mathbb{R}^n\text{, denoted by } \lVert \mathbf{x} \rVert \\ \text{for } \mathbf{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n\text{, is defined as} \\ \lVert \mathbf{x} \rVert := \left( \sum_{j=1}^{n} x_j^2 \right)^{1/2} = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}
Definition 2. Distance
§4.1
The distance between two pointsx=(x1,…,xn), y=(y1,…,yn)∈Rn is defined as∥x−y∥=∥(x1−y1,…,xn−yn)∥=(∑i=1n(xi−yi)2)1/2=(x1−y1)2+(x2−y2)2+⋯+(xn−yn)2\text{The distance between two points} \\ \mathbf{x} = (x_1, \ldots, x_n),\ \mathbf{y} = (y_1, \ldots, y_n) \in \mathbb{R}^n \text{ is defined as} \\ \begin{aligned} \lVert \mathbf{x} - \mathbf{y} \rVert &= \lVert (x_1 - y_1, \ldots, x_n - y_n) \rVert \\ &= \left( \sum_{i=1}^{n} (x_i - y_i)^2 \right)^{1/2} \\ &= \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2} \end{aligned}
Definition 3. Dot Product / Inner Product
§4.1
For two vectors x=(x1,…,xn), y=(y1,…,yn)∈Rn,the dot product, denoted by ⟨x,y⟩, is defined as⟨x,y⟩:=∑j=1nxjyj=x1y1+⋯+xnynRecall, the dot product is linear in each variable:for x,y,z∈Rn, t,s∈R,⟨tx+sy,z⟩=t⟨x,z⟩+s⟨y,z⟩⟨x,ty+sz⟩=t⟨x,y⟩+s⟨x,z⟩Also, it is symmetric:for x=(x1,…,xn), y=(y1,…,yn)∈Rn,⟨x,y⟩=∑j=1nxjyj=∑j=1nyjxj=⟨y,x⟩Also note, for x=(x1,…,xn)∈Rn,⟨x,x⟩=∑j=1nxj⋅xj=∑j=1nxj2=∥x∥2\text{For two vectors } \mathbf{x} = (x_1, \ldots, x_n),\ \mathbf{y} = (y_1, \ldots, y_n) \in \mathbb{R}^n\text{,} \\ \text{the dot product, denoted by } \langle \mathbf{x}, \mathbf{y} \rangle \text{, is defined as} \\ \langle \mathbf{x}, \mathbf{y} \rangle := \sum_{j=1}^{n} x_j y_j = x_1 y_1 + \cdots + x_n y_n \\[1em] \text{Recall, the dot product is linear in each variable:} \\ \text{for } \mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^n,\ t, s \in \mathbb{R}\text{,} \\ \begin{aligned} \langle t\mathbf{x} + s\mathbf{y}, \mathbf{z} \rangle &= t \langle \mathbf{x}, \mathbf{z} \rangle + s \langle \mathbf{y}, \mathbf{z} \rangle \\ \langle \mathbf{x}, t\mathbf{y} + s\mathbf{z} \rangle &= t \langle \mathbf{x}, \mathbf{y} \rangle + s \langle \mathbf{x}, \mathbf{z} \rangle \end{aligned} \\[1em] \text{Also, it is symmetric:} \\ \text{for } \mathbf{x} = (x_1, \ldots, x_n),\ \mathbf{y} = (y_1, \ldots, y_n) \in \mathbb{R}^n\text{,} \\ \langle \mathbf{x}, \mathbf{y} \rangle = \sum_{j=1}^{n} x_j y_j = \sum_{j=1}^{n} y_j x_j = \langle \mathbf{y}, \mathbf{x} \rangle \\[1em] \text{Also note, for } \mathbf{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n\text{,} \\ \langle \mathbf{x}, \mathbf{x} \rangle = \sum_{j=1}^{n} x_j \cdot x_j = \sum_{j=1}^{n} x_j^2 = \lVert \mathbf{x} \rVert^2
Definition 4. Collinear
§4.1
Let x,y∈Rn.x and y are collinear iff  ∃t∈R s.t. y=tx.\text{Let } \mathbf{x}, \mathbf{y} \in \mathbb{R}^n. \\ \mathbf{x} \text{ and } \mathbf{y} \text{ are collinear iff } \ \exists t \in \mathbb{R} \text{ s.t. } \mathbf{y} = t \mathbf{x}.
Definition 5. Orthonormal
§4.1
A set {v1,…,vn}⊆Rn of vectorsis orthonormal iff⟨vi,vj⟩=δij:={1if i=j0if i≠j.Note ∥vj∥=⟨vj,vj⟩=1=1\text{A set } \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ of vectors} \\ \text{is orthonormal iff} \\ \langle \mathbf{v}_i, \mathbf{v}_j \rangle = \delta_{ij} := \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases} \\[1em] \text{Note } \lVert \mathbf{v}_j \rVert = \sqrt{\langle \mathbf{v}_j, \mathbf{v}_j \rangle} = \sqrt{1} = 1
Definition 6. Orthonormal Basis (ONB)
§4.1
A set S={v1,…,vn}⊆Rn of vectorsis an orthonormal basis (ONB) if Sis orthonormal and span⁡(S)=Rn.\text{A set } S = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ of vectors} \\ \text{is an orthonormal basis (ONB) if } S \\ \text{is orthonormal and } \operatorname{span}(S) = \mathbb{R}^n.
Definition 7. Linearly Independent
§4.1
A set S={v1,…,vn}⊆Rn of vectorsis linearly independent iff for a1,…,an∈R,∑j=1najvj=0  ⟹  aj=0 ∀j.\text{A set } S = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ of vectors} \\ \text{is linearly independent iff for } a_1, \ldots, a_n \in \mathbb{R}, \\ \sum_{j=1}^{n} a_j \mathbf{v}_j = \mathbf{0} \implies a_j = 0 \ \forall j.
Definition 8. Basis
§4.1
A set S={v1,…,vn}⊆Rn of vectorsis a basis iff S is linearly independent‾and span⁡(S)=Rn.\text{A set } S = \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ of vectors} \\ \text{is a basis iff } S \text{ is } \underline{\href{/notes/real-analysis/definitions#definition-7}{\text{linearly independent}}} \\ \text{and } \operatorname{span}(S) = \mathbb{R}^n.
Definition 9. Convergence of a Sequence
§4.2
A sequence of points (xk)⊆Rnconverges to a point a∈Rn if∀ε>0, ∃N∈N, ∀k≥N, ∥xk−a∥<ε.In this case, we notate this by lim⁡k→∞xk=a.\text{A sequence of points } (\mathbf{x}_k) \subseteq \mathbb{R}^n \\ \text{converges to a point } \mathbf{a} \in \mathbb{R}^n \text{ if} \\ \forall \varepsilon > 0,\ \exists N \in \mathbb{N},\ \forall k \geq N,\ \lVert \mathbf{x}_k - \mathbf{a} \rVert < \varepsilon. \\ \text{In this case, we notate this by } \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a}.
Definition 10. Cauchy Sequence
§4.2
A sequence (xk)⊆Rn is Cauchy if∀ε>0, ∃N∈N, ∀k,l≥N, ∥xk−xl∥<ε.\text{A sequence } (\mathbf{x}_k) \subseteq \mathbb{R}^n \text{ is Cauchy if} \\ \forall \varepsilon > 0,\ \exists N \in \mathbb{N},\ \forall k, l \geq N,\ \lVert \mathbf{x}_k - \mathbf{x}_l \rVert < \varepsilon.
Definition 11. Complete
§4.2
A set S⊆Rn is complete if everyCauchy sequence‾ in S converges‾ to a point in S.(xk)⊆S, (xk) Cauchy  ⟹  ∃a∈Rn, lim⁡k→∞xk=a∈S\text{A set } S \subseteq \mathbb{R}^n \text{ is complete if every} \\ \underline{\href{/notes/real-analysis/definitions#definition-10}{\text{Cauchy sequence}}} \text{ in } S \text{ } \underline{\href{/notes/real-analysis/definitions#definition-9}{\text{converges}}} \text{ to a point in } S. \\[0.5em] (\mathbf{x}_k) \subseteq S,\ (\mathbf{x}_k) \text{ Cauchy} \implies \exists \mathbf{a} \in \mathbb{R}^n,\ \lim_{k \to \infty} \mathbf{x}_k = \mathbf{a} \in S
Definition 12. Limit Point
§4.3
A point x∈Rn is a limit point of a set A⊆Rnif ∃(ak)⊆A such that lim⁡k→∞ak=x.\text{A point } \mathbf{x} \in \mathbb{R}^n \text{ is a limit point of a set } A \subseteq \mathbb{R}^n \\ \text{if } \exists (\mathbf{a}_k) \subseteq A \text{ such that } \lim_{k \to \infty} \mathbf{a}_k = \mathbf{x}.
Definition 13. Closed
§4.3
A set A⊆Rn is closed if it containsall of its limit points‾.In other words, Closed means {limit pts of A}⊆A.Note for any a∈A, we may consider the constantsequence ak≡a ∀k to see a∈{limit pts of A}.That is, {limit pts of A}⊇A always holds.So A is closed iff A={limit pts of A}, and to showA is closed we only need to show all limit points are in A.\text{A set } A \subseteq \mathbb{R}^n \text{ is closed if it contains} \\ \text{all of its \underline{\href{/notes/real-analysis/definitions#definition-12}{\text{limit points}}}.} \\[1em] \text{In other words,} \text{ Closed means } \{\text{limit pts of } A\} \subseteq A. \\[1em] \text{Note for any } \mathbf{a} \in A, \text{ we may consider the constant} \\ \text{sequence } \mathbf{a}_k \equiv \mathbf{a}\ \forall k \text{ to see } \mathbf{a} \in \{\text{limit pts of } A\}. \\ \text{That is, } \{\text{limit pts of } A\} \supseteq A \text{ always holds.} \\ \text{So } A \text{ is closed iff } A = \{\text{limit pts of } A\}, \text{ and to show} \\ A \text{ is closed we only need to show all limit points are in } A.
Definition 14. Closure
§4.3
Let A⊆Rn. The closure of A is the set A‾consisting of all of its limit points‾.Note A closed‾  ⟺  A‾⊆A.\text{Let } A \subseteq \mathbb{R}^n. \text{ The closure of } A \text{ is the set } \overline{A} \\ \text{consisting of all of its \underline{\href{/notes/real-analysis/definitions#definition-12}{\text{limit points}}}.} \\[1em] \text{Note } A \text{ \underline{\href{/notes/real-analysis/definitions#definition-13}{\text{closed}}}} \iff \overline{A} \subseteq A.
Definition 15. Dense
§4.3
A set A is dense in B if A⊆B⊆A‾.e.g. Q is dense in R, since Q‾=R.\text{A set } A \text{ is dense in } B \text{ if } A \subseteq B \subseteq \overline{A}. \\[1em] \text{e.g. } \mathbb{Q} \text{ is dense in } \mathbb{R}, \text{ since } \overline{\mathbb{Q}} = \mathbb{R}.
Definition 16. Ball
§4.3
The ball about a∈Rn of radius r is the setBr(a):={x∈Rn∣∥x−a∥<r}.\text{The ball about } \mathbf{a} \in \mathbb{R}^n \text{ of radius } r \text{ is the set} \\ B_r(\mathbf{a}) := \{ \mathbf{x} \in \mathbb{R}^n \mid \lVert \mathbf{x} - \mathbf{a} \rVert < r \}.
Definition 17. Open
§4.3
A set U⊆Rn is open if∀a∈U, ∃r=r(a)>0, Br(a)‾⊆U.\text{A set } U \subseteq \mathbb{R}^n \text{ is open if} \\ \forall \mathbf{a} \in U,\ \exists r = r(\mathbf{a}) > 0,\ \underline{\href{/notes/real-analysis/definitions#definition-16}{\text{$B_r(\mathbf{a})$}}} \subseteq U.
Definition 18. Interior
§4.3
The interior of a set X⊆Rn is the largestopen‾ set contained in X, and is denoted int⁡(X).\text{The interior of a set } X \subseteq \mathbb{R}^n \text{ is the largest} \\ \text{\underline{\href{/notes/real-analysis/definitions#definition-17}{\text{open}}} set contained in } X, \text{ and is denoted } \operatorname{int}(X).
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