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Convergence of Sequences in Normed Spaces 📂Banach Space

Convergence of Sequences in Normed Spaces

Definition

Let’s call (X,∥⋅∥)(X, \left\| \cdot \right\|) as a normed space. For a sequence {xn}\left\{ x_{n} \right\} of XX,

lim⁡n→∞∥x−xn∥=0,x∈X \lim \limits_{n \to \infty} \left\| x - x_{n} \right\| = 0,\quad x\in X

it is said to converge to xx if it satisfies the following condition, and it is represented as follows.

xn→x as n→∞orx=lim⁡n→∞xn x_{n} \to x \text { as } n \to \infty \quad \text{or} \quad x=\lim \limits_{n\to\infty}x_{n}

Explanation

To define convergence, a distance is needed, but since distance can be naturally defined as d(x,y)=∥x−y∥d(x,y)=\left\| x - y \right\| in normed spaces (../1840), it’s similar to the definition in metric spaces except that the metric is replaced with a norm.

If for every ϵ>0\epsilon >0, there exists a natural number N∈NN\in \mathbb{N} satisfying the equation below, then the sequence {xn}\left\{ x_{n} \right\} is said to converge to xx.

∥x−xn∥<ϵ∀n≥N \left\| x - x_{n} \right\|<\epsilon \quad \forall n \ge N

Compared to weak convergence, it is also referred to as strongly converging.

xn converges to x= xn converges in norm to x= xn converges strongly to x \begin{align*} & x_{n} \text{ converges to } x \\ =&\ x_{n} \text{ converges in norm to } x \\ =&\ x_{n} \text{ converges strongly to } x \end{align*}