##  [Seminorm](/seminorm-1) 

 Definition

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

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

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

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

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

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

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

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

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.