#sequences#convergence#componentwise#lemma86

Unit 1 · 8.1 · Vectors, norms & sequences

Sequences in ℝᵈ

A vector sequence converges if and only if every component converges — equivalently, if the distance $‖ x_{k} - y ‖_{2}$ goes to $0$ .

Chapters

Formulas at a glance · Vectors, norms & sequences

Scalar product & Euclidean norm

u \cdot v = \sum_{i = 1}^{n} u_{i} v_{i}, ‖ v ‖_{2} = \sqrt{v \cdot v}, | u \cdot v | \leq ‖ u ‖_{2} ‖ v ‖_{2}

Norms & unit balls

‖ v ‖_{\infty} \leq ‖ v ‖_{2} \leq \sqrt{d} ‖ v ‖_{\infty}, \frac{1}{\sqrt{d}} ‖ v ‖_{1} \leq ‖ v ‖_{2} \leq ‖ v ‖_{1}

Sequences in ℝᵈ

x_{k} \to y ⟺ ‖ x_{k} - y ‖_{2} \to 0