Catalog of theorems from Real Analysis alongside proofs for each.

Theorem 1. Cauchy-Schwarz Inequality
§4.1
For x,y∈Rn, ∣⟨x,y⟩∣≤∥x∥∥y∥Equality holds iff x and y are collinear.\text{For } \mathbf{x}, \mathbf{y} \in \mathbb{R}^n, \ |\langle x, y \rangle | \leq \| \mathbf{x} \| \| \mathbf{y} \| \\ \text{Equality holds iff } \mathbf{x} \text{ and } \mathbf{y} \text{ are collinear}.

Lemma 2. Properties of Orthonormal Sets
§4.1
Let {v1,…,vn}⊆Rn be an orthonormal set.(a) For aj∈R, ∥∑j=1najvj∥=(∑j=1naj2)1/2(b) Any orthonormal set in Rn is linearly independent.(c) An ONB in Rn is a basis and has exactly n elements.\text{Let } \{\mathbf{v}_1, \ldots, \mathbf{v}_n\} \subseteq \mathbb{R}^n \text{ be an orthonormal set.} \\[0.5em] \text{(a) For } a_j \in \mathbb{R},\ \left\lVert \sum_{j=1}^{n} a_j \mathbf{v}_j \right\rVert = \left( \sum_{j=1}^{n} a_j^2 \right)^{1/2} \\[0.5em] \text{(b) Any orthonormal set in } \mathbb{R}^n \text{ is linearly independent.} \\[0.5em] \text{(c) An ONB in } \mathbb{R}^n \text{ is a basis and has exactly } n \text{ elements.}

Lemma 3. Triangle Inequality
§4.1
For all x,y∈Rn, ∥x+y∥≤∥x∥+∥y∥.Moreover, the equality holds iffx=0 or ∃c≥0, y=cx.\text{For all } \mathbf{x}, \mathbf{y} \in \mathbb{R}^n,\ \lVert \mathbf{x} + \mathbf{y} \rVert \leq \lVert \mathbf{x} \rVert + \lVert \mathbf{y} \rVert. \\ \text{Moreover, the equality holds iff} \\ \mathbf{x} = \mathbf{0} \text{ or } \exists c \geq 0,\ \mathbf{y} = c\mathbf{x}.

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