A seminorm p on a vector space is a nonnegative functional satisfying absolute homogeneity p(αv)=|α|p(v) and subadditivity p(u+v)≤p(u)+p(v), but it may vanish on nonzero vectors (its kernel can be nontrivial), so positive definiteness can fail.

Principle

Organizing idea: a seminorm measures size up to a possible null subspace; it preserves scaling and subadditivity but allows a nontrivial kernel, making it suitable to generate locally convex topologies via families of seminorms.

Demonstration

Example: on the space of continuous functions C(R), define p(f)=|f(0)|. p is a seminorm: homogeneous and subadditive, but p(f)=0 for any f vanishing at 0, so nonzero functions can have zero seminorm. Families of such seminorms define standard topologies on spaces of smooth functions or distributions.

Misapplication

Using a seminorm as if it were a norm when invertibility or uniqueness is needed; for example treating p(f)=0 as implying f=0 leads to false deductions. Another misuse is ignoring the kernel when forming quotient spaces where the kernel must be modded out.

Consequence

Seminorms allow flexible topological constructions: finite or directed families of seminorms generate locally convex topologies, duality frameworks and distribution spaces. Passing to the quotient by the kernel converts a seminorm into a genuine norm.

Reversal

The reversal is a norm, which is a seminorm with trivial kernel (p(v)=0 only for v=0). Where a seminorm fails to detect certain directions, a norm fully detects nonzero vectors.

Boundary

Defined on vector spaces and requiring homogeneity and subadditivity; it does not require positive definiteness. Seminorms are distinct from quasinorms (which relax the triangle inequality) and from gauges that may lack homogeneity.

Semantic Tension

Tension appears between seminorms and norms, and between single seminorms and families: a single seminorm gives limited separation, whereas a separating family of seminorms can induce a Hausdorff topology. One must distinguish seminorm kernels from genuine null vectors.

Synthesis

A seminorm is a size measure on a vector space that respects scaling and subadditivity but may vanish on a subspace; used in families it generates the locally convex topologies central to functional analysis.