Analysis MOC

Seminormed vector space

A seminormed vector space (𝑉,𝕂,‖ ⋅‖) is a Vector space over a Subfield 𝕂 of ℂ equipped with a seminorm ‖ ⋅‖ :𝑉 →ℝ, a weakening of a norm satisfying the following conditions for any 𝑥,𝑦 ∈𝑉 and 𝛼 ∈𝕂 #m/def/anal/vec

1. Absolute homogeneity: ‖𝛼𝑥‖ =|𝛼|‖𝑥‖
2. Triangle inequality: ‖𝑥 +𝑦‖ ≤‖𝑥‖ +‖𝑦‖

whence follows

3. Nonnegativity: ‖𝑥‖ ≥0

By strengthening nonnegativity to positive-definiteness the seminorm becomes a full norm.

Properties

1. A seminorm induces a normed quotient

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