A function p: V → [0,∞) on a vector space V that is positively homogeneous (p(αv)=|α| p(v) for scalars α) and subadditive (p(v+w) ≤ p(v)+p(w)); unlike a norm, p(v)=0 may hold for some nonzero v.

Principle

A seminorm measures magnitude up to a kernel: ker p = {v | p(v)=0} is a linear subspace, and p descends to a genuine norm on the quotient space V/ker p; families of seminorms generate locally convex topologies.

Demonstration

Example on C([0,1]): p(f)=|f(0)| is a seminorm — it is homogeneous and subadditive but vanishes on all functions that are zero at 0, a nontrivial kernel.

Misapplication

Treating a seminorm as a norm without quotienting by its kernel (for instance using p to define a metric on V directly) or expecting invertibility/injectivity conclusions that require definiteness.

Consequence

Seminorms allow construction of topologies suited to functional analysis (locally convex spaces), permit continuity and boundedness tests, and enable passage to norms by quotienting or combining families of seminorms.

Reversal

The reverse is a norm, which additionally requires trivial kernel; moving from seminorm to norm typically requires modding out the kernel or strengthening positivity.

Boundary

Applies to vector spaces over R or C; a seminorm is not required to generate a Hausdorff topology unless the separating family condition holds (intersection of kernels = {0}).

Semantic Tension

Tension between seminorm and arbitrary subadditive positively homogeneous gauges that may lack linearity properties; also between seminorm and semi-inner-product constructions that impose extra structure.

Synthesis

A seminorm is a relaxed norm that measures size while allowing a linear kernel of indistinguishable vectors; it is the basic building block for locally convex topologies and for forming norms on quotient spaces.