LemmaStatement: Literature-sourcedProof: AI-generated

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Vector addition and scalar multiplication are continuous in a normed space

Statement

Let $V$ be a normed space over $R$ or $C$ . Then

$+ : V \times V \to V, (x, y) \mapsto x + y,$

and

$m : K \times V \to V, (λ, x) \mapsto λ x,$

are continuous for the product topology on the domain and the norm topology on the target.

Facts & Assumptions

Given: Scalars $λ, α \in K$ and vectors $x, y, h, k \in V$ .

[L1]

The scalar field on this page is either $R$ with its usual absolute value or $C$ with its modulus, and the same norm estimates are valid in both cases (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Complex norm and complex normed space). The norm topology is that of $d (x, y) = ∥ x - y ∥$ ; in the complex case this is the same metric after realification (Realification preserves a complex norm and its metric).

[L2]

The norm is $1$ -Lipschitz for its metric, so in particular $∥ u + v ∥ \leq ∥ u ∥ + ∥ v ∥$ for all vectors and $∣ ∥ u ∥ - ∥ v ∥ ∣ \leq ∥ u - v ∥$ (The reverse triangle inequality in a normed space).

Proof

technique · direct

1.1L2

For addition, $∥ (x + h) + (y + k) - (x + y) ∥ = ∥ h + k ∥ \leq ∥ h ∥ + ∥ k ∥$ by [L2].

1.2L1L2algebra

For scalar multiplication, $(λ + α) (x + h) - λ x = λ h + α x + α h$ , so $∥ (λ + α) (x + h) - λ x ∥ \leq ∣ λ ∣ ∥ h ∥ + ∣ α ∣ ∥ x ∥ + ∣ α ∣ ∥ h ∥$ by the triangle inequality and absolute homogeneity from [L1] and [L2].

2.1step 1.1given

Given $ε > 0$ , step 1.1 shows that if $∥ h ∥ < ε / 2$ and $∥ k ∥ < ε / 2$ , then $(x + h) + (y + k)$ lies in the $ε$ -ball around $x + y$ ; this is continuity of addition at $(x, y)$ .

2.2step 1.2givenalgebra

For scalar multiplication at $(λ, x)$ , first require $∣ α ∣ < 1$ and then choose $δ > 0$ so that $∣ λ ∣ δ + δ (∥ x ∥ + δ) < ε$ ; step 1.2 then gives $∥ (λ + α) (x + h) - λ x ∥ < ε$ whenever $∣ α ∣ < δ$ and $∥ h ∥ < δ$ .

3.1step 2.1step 2.2∎

Since the point $(x, y)$ was arbitrary, step 2.1 proves continuity of addition everywhere, and step 2.2 proves continuity of scalar multiplication everywhere.

Vector addition and scalar multiplication are continuous in a normed space

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources