A seminorm induces a normed quotient

Let $(𝑉, 𝕂, ‖ \cdot ‖)$ be a Seminormed vector space and let

N = {𝑥 \in 𝑉 : ‖ 𝑥 ‖ = 0}

Then the Quotient vector space $𝑉 / 𝑁$ carries a unique norm such that #m/def/anal/vec

‖ 𝑥 + 𝑁 ‖ = ‖ 𝑥 ‖

Proof

Note that for any $𝑦 \in 𝑁$ , $‖ 𝑥 + 𝑦 ‖ \leq ‖ 𝑥 ‖ + ‖ 𝑦 ‖ = ‖ 𝑥 ‖$ . Now

‖ 𝛼 (𝑥 + 𝑁) ‖ = ‖ 𝛼 𝑥 + 𝑁 ‖ = ‖ 𝛼 𝑥 ‖ = ‖ 𝛼 ‖ ‖ 𝑥 ‖ = | 𝛼 | ‖ 𝑥 + 𝑁 ‖

‖ (𝑥 + 𝑁) + (𝑦 + 𝑁) ‖ = ‖ 𝑥 + 𝑦 + 𝑁 ‖ = ‖ 𝑥 + 𝑦 ‖ \leq ‖ 𝑥 ‖ + ‖ 𝑦 ‖ = ‖ 𝑥 + 𝑁 ‖ + ‖ 𝑦 + 𝑁 ‖

proving the triangle inequality. Finally $‖ 𝑥 + 𝑁 ‖ = ‖ 𝑥 ‖ = 0$ implies $𝑥 = 𝑁$ , so $𝑥 + 𝑁 = 𝑁$ , thus we have positive definiteness.

#state/tidy | #lang/en | #SemBr