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.
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