Convergence in normed spaces

Let $(X,\|\cdot\|)$ be a normed vector space .

A sequence $(x_n)$ in $X$ converges to $x\in X$ (in norm) if

Equivalently: for every $\varepsilon>0$ there exists $N$ such that $\|x_n-x\|<\varepsilon$ for all $n\ge N$.

Context. By the metric induced by a norm , this is exactly convergence in the associated metric space .

Examples:

• In $\mathbb{R}^2$ with $\|\cdot\|_2$, the sequence $x_n=(1/n,0)$ converges to $(0,0)$.
• In $C([0,1])$ with $\|\cdot\|_\infty$, the functions $f_n(t)=t/n$ satisfy $\|f_n-0\|_\infty=1/n\to 0$, so $f_n\to 0$.
• A sequence may converge under one norm and not under another in infinite-dimensional spaces (choice of norm matters).