Knowlpedia

Algebra of limits in normed spaces

Limits commute with addition and scalar multiplication

Proposition. Let (X,∥⋅∥)(X,\|\cdot\|) be a normed space.

  1. If xn→xx_n\to x and yn→yy_n\to y, then xn+yn→x+yx_n+y_n\to x+y.
  2. If xn→xx_n\to x and αn→α\alpha_n\to \alpha (in R\mathbb{R} or C\mathbb{C}), then αnxn→αx\alpha_n x_n\to \alpha x.
Remarks

Context. These are the basic "limit laws" for sequences in normed linear settings.

Proof sketch.

  1. By the triangle inequality,
∥(xn+yn)−(x+y)∥≤∥xn−x∥+∥yn−y∥→0.\|(x_n+y_n)-(x+y)\|\le \|x_n-x\|+\|y_n-y\|\to 0.
  1. Write
∥αnxn−αx∥≤∥αn(xn−x)∥+∥(αn−α)x∥=∣αn∣ ∥xn−x∥+∣αn−α∣ ∥x∥.\|\alpha_n x_n-\alpha x\|\le \|\alpha_n(x_n-x)\|+\|(\alpha_n-\alpha)x\| =|\alpha_n|\,\|x_n-x\|+|\alpha_n-\alpha|\,\|x\|.

Use that (αn)(\alpha_n) is bounded and both factors tend to 00.