1 Definition and basic properties

A seminorm on a vector space \(V\) over a field (typically \(\mathbb{R}\) or \(\mathbb{C}\)) is a function \(p:V\to[0,\infty)\) that is absolutely homogeneous and subadditive. The distinguishing feature from a norm is that it may assign value \(0\) to nonzero vectors, so it measures size only up to some “invisible directions.”

1.1 Absolute homogeneity and the triangle inequality

The defining axioms are:

  • Absolute homogeneity: for every scalar \(\alpha\) and vector \(x\in V\),

\[

p(\alpha x)=\alpha\,p(x).

\]

\[ p(x+y)\le p(x)+p(y). \] From these, one obtains standard consequences such as \(p(0)=0\) and \(p(-x)=p(x)\).

1.2 Null space (kernel) of a seminorm

The set \[ \ker p=\{x\in V: p(x)=0\} \] is a linear subspace of \(V\). It contains \(0\), is closed under addition because if \(p(x)=p(y)=0\) then \(p(x+y)\le 0\), and is closed under scalar multiplication because absolute homogeneity preserves the zero value. The seminorm becomes a norm precisely when \(\ker p=\{0\}\).

1.3 Seminorms versus norms

Every norm is a seminorm, but not conversely. The only relaxation is the permitted existence of nonzero vectors of seminorm \(0\). Geometrically, a seminorm defines “distance” in a quotient-like manner: vectors differing by an element of \(\ker p\) cannot be distinguished by \(p\).

1.4 Examples from common function spaces

Common sources of seminorms include:

- Derivative-based seminorms. On spaces of differentiable functions, a quantity like \(p(f)=\|f'\|\) (with \(\|\cdot\|\) an underlying norm) vanishes on constant functions, so it is a seminorm.
  • Sobolev-type seminorms. In Sobolev spaces, the “seminorm” part measures derivatives but not the average or lower-order terms; it vanishes on polynomials of bounded degree depending on the order.
  • Energy forms. In variational problems, quadratic forms often define seminorms when they ignore certain rigid motions or null modes.

2 Algebraic and geometric structure

A seminorm can be interpreted through multiple equivalent viewpoints: as a subadditive magnitude function, as a tool to build pseudo-metrics, or as a device encoding convexity.

2.1 Subadditivity viewpoint

Subadditivity is the triangle inequality itself. It implies that \(p\) behaves like a measure of “cost” under addition: combining vectors cannot increase the seminorm beyond the sum of individual costs.

2.2 Level sets and induced pseudo-metrics

For any \(r>0\), the sublevel set \[ \{x\in V: p(x)<r\} \] is absorbing and balanced in the direction sense typical of seminorm balls. Using a single seminorm, one can define a pseudo-metric \[ d_p(x,y)=p(x-y), \] which satisfies symmetry and the triangle inequality, but may assign distance \(0\) to distinct points when \(x-y\in\ker p\).

2.3 Convexity interpretation

The unit ball \(B_p=\{x: p(x)\le 1\}\) is convex. More generally, because \(p\) is subadditive and homogeneous, the sets \(\{x: p(x)\le r\}\) are convex for all \(r\ge 0\). This links seminorms to convex geometry and underlies many results about locally convex topologies.

2.4 Equivalence under scaling

If one multiplies a seminorm by a positive constant \(c\), the essential structure remains: \(cp\) is again a seminorm, and it induces the same null space and the same notion of which sequences have seminorm going to zero (up to rescaling). Two seminorms can also be topologically equivalent even when they are not simple scalar multiples, provided they induce the same local behavior around zero.

3 Seminorm-induced topology

A single seminorm yields a pseudo-metric and therefore a canonical topology, often non-Hausdorff when the kernel is nontrivial.

3.1 Pseudo-metrics from a single seminorm

With \(d_p(x,y)=p(x-y)\), the balls \[ B_{d_p}(x,\varepsilon)=\{y: p(y-x)&lt;\varepsilon\} \] define neighborhoods. Distinct points may be topologically indistinguishable if they differ by an element in \(\ker p\), reflecting the failure of separation in the underlying topology.

3.2 Convergence and Cauchy sequences

  • A sequence \((x_n)\) converges to \(x\) in the seminorm topology iff \(p(x_n-x)\to 0\).
  • A sequence is Cauchy iff \(p(x_n-x_m)\to 0\) as \(n,m\to\infty\).

Because \(p\) may vanish on nonzero vectors, Cauchy sequences need not converge to a unique limit unless the topology is Hausdorff.

3.3 Continuity in seminorm topologies

Let \(T:V\to W\) be a map between vector spaces with seminorms \(p\) on \(V\) and \(q\) on \(W\). Continuity at a point can be expressed by how \(q(T(x)-T(x_0))\) depends on \(p(x-x_0)\). For linear maps, continuity is equivalent to a uniform seminorm estimate (discussed in later sections).

3.4 Completeness considerations

Completeness relative to a seminorm topology is typically framed using Cauchy sequences modulo the kernel. Even when the pseudo-metric space is not Hausdorff, one can often obtain a Hausdorff complete space by passing to a quotient by \(\ker p\), where the induced object becomes a true normed space.

4 Families of seminorms and locally convex spaces

Many spaces in analysis are naturally described not by a single seminorm but by a whole family, producing a locally convex topology.

4.1 Defining locally convex topology via seminorm families

Given a family \(\{p_i\}_{i\in I}\) of seminorms on \(V\), one defines the locally convex topology as the coarsest topology in which each \(p_i\) is continuous. A neighborhood basis at zero consists of finite intersections of sets where all finitely many seminorms are small.

4.2 Fundamental neighborhoods of the zero vector

Typical basic neighborhoods of \(0\) have the form \[ U=\{x\in V: p_{i_1}(x)<\varepsilon_1,\dots,p_{i_n}(x)<\varepsilon_n\} \] for some finite subset \(\{i_1,\dots,i_n\}\subset I\) and positive numbers \(\varepsilon_k\). Translating these sets gives neighborhoods around arbitrary points.

4.3 Separation and the role of the intersection of kernels

The resulting topology is Hausdorff exactly when the family separates points, i.e. \[ \bigcap_{i\in I}\ker p_i=\{0\}. \] If the intersection is nontrivial, distinct vectors can not be distinguished by any seminorm in the family, producing a non-Hausdorff topological vector space.

4.4 Topological vector space consequences

With addition and scalar multiplication continuous under the seminorm-generated topology, the structure becomes a topological vector space. The local convexity comes from convexity of seminorm balls, making it possible to use tools from convex analysis, such as separation theorems, in settings like locally convex spaces.

5 Quotients and norm induced on factor spaces

A core reason seminorms are useful is that they naturally lead to quotient constructions where the seminorm becomes a genuine norm.

5.1 Construction of the quotient space by the kernel

Given \(p\) on \(V\), form the quotient vector space \[ V/\ker p. \] Elements are equivalence classes \(x+\ker p\), where \(x\sim y\) iff \(p(x-y)=0\). Vectors differing by something invisible to \(p\) become identified.

5.2 Induced norm and well-definedness

Define \[

\|x+\ker p\| = p(x).

\] This is well-defined: if \(x-y\in\ker p\), then \(p(x)=p(y)\). The induced function is a norm because its kernel in the quotient is trivial.

5.3 Universal property of the quotient with respect to the seminorm

The quotient map \(\pi:V\to V/\ker p\) has a universal characterization: any linear map from \(V\) that annihilates \(\ker p\) factors uniquely through \(\pi\). Moreover, constructions based on \(p\) (such as continuity or operator bounds) can be transported to the quotient where the geometry is properly normed.

5.4 Examples of quotienting to obtain norms

  • In derivative seminorms, quotienting by constant functions yields a norm on equivalence classes of functions modulo constants.
  • In energy methods for PDE, quotienting by the space of null modes produces a norm that measures “true energy” rather than rigid degrees of freedom.

6 Continuity and boundedness of linear maps

Seminorms provide a precise language for when linear maps behave regularly with respect to the topology.

6.1 Operator seminorms and boundedness

If \(T:V\to W\) is linear and \(p\) and \(q\) are seminorms, a standard quantity is \[

\|T\|_{p\to q}=\sup_{x\neq 0}\frac{q(Tx)}{p(x)},

\] interpreted appropriately when \(p(x)=0\). When this supremum is finite and consistent with kernels, \(T\) is bounded between the seminormed structures.

6.2 Continuity criteria using seminorms

For linear maps, continuity can be characterized by inequalities of the form \[ q(Tx)\le C\,p(x)\quad \text{for all }x\in V \] for some \(C\ge 0\). In families of seminorms, continuity means that each target seminorm \(q_j\) is controlled by finitely many domain seminorms in a suitable way.

6.3 Induced maps on quotient spaces

If \(T\) maps \(\ker p\) into \(\ker q\), then \(T\) induces a linear map \[ \tilde{T}:V/\ker p \to W/\ker q. \] When passing to quotients where the seminorms become norms, this induced map inherits boundedness properties compatible with the operator norms of the quotient spaces.

6.4 Seminorm estimates and inequalities

Many arguments reduce to bounding seminorms via algebraic inequalities. Examples include:

  • chaining estimates through compositions of linear maps,
  • using subadditivity to control differences \(T(x)-T(y)=T(x-y)\),
  • deriving inequalities for seminorms of products or derivatives when seminorms are defined from those operations.

7 Duality and polar constructions (seminorm perspective)

Duality links seminorms with families of continuous linear functionals, often through polar sets and support functions.

7.1 Dual seminorms arising from continuous linear functionals

Given a normed or seminormed setting, continuous linear functionals can be used to produce another seminorm (or norm on an appropriate quotient). Typically, one defines a dual quantity by taking the supremum of \(f(x)\) over functionals \(f\) that satisfy a normalization condition, thereby measuring how large \(x\) is “from the viewpoint of linear tests.”

7.2 Polar sets and relationship to seminorms

For a subset \(A\) in a vector space, its polar involves all linear functionals that are bounded by a threshold on \(A\). Seminorm balls have polars that are naturally described in terms of the dual space and the normalized constraints on functionals. This dual relationship is a central mechanism behind separation and representation results.

7.3 Support function viewpoint in convex analysis

For convex sets, support functions encode the maximal pairing in each direction. When sets are defined by seminorm constraints, the support function can often be reexpressed in terms of dual seminorms, tying seminorm geometry to convex optimization and separation.

7.4 Norming sets and separation by functionals

A family of functionals is said to norm a space if it controls the seminorm or separates points. In locally convex spaces, separation properties can be interpreted as the existence of enough continuous linear functionals so that kernels of seminorms correspond to common nullities of certain functional families.

8 Constructing seminorms in analysis

Seminorms often arise by design from analytic quantities—measuring growth, smoothness, or energy—while deliberately ignoring certain components.

8.1 Using derivatives (e.g., smoothness seminorms)

In function spaces, a common approach is to measure derivatives up to a certain order in a specified way. Such constructions may vanish on lower-degree components (for instance, derivative-based seminorms do not detect constants when only first derivatives are used), producing a seminorm rather than a norm.

8.2 Weighted seminorms and growth conditions

Weights allow seminorms to distinguish behavior at infinity or near singularities. Multiplying by a weight function changes which functions are “small” in the seminorm sense and can enforce decay or control in specific regions, while still possibly leaving a kernel of functions that the weighted measurement fails to detect.

8.3 Energy-type seminorms in PDE and variational settings

In PDE and the calculus of variations, one frequently defines seminorms from quadratic forms or bilinear forms that represent energy. When the form is degenerate—due to symmetries or constraints—the corresponding functional is a seminorm. Quotienting by the null space yields the physically or mathematically meaningful norm.

8.4 Seminorms on spaces of distributions (overview-level)

Distributions are measured via how they act on test functions. Seminorm-like control can be organized by bounding the action of a distribution on families of test functions with prescribed bounds on derivatives. While the precise construction depends on the chosen framework, the overarching idea is that “size” is assessed through boundedness against structured probes, frequently leading to families of seminorms rather than a single one.

9 Operations on seminorms

Seminorms can be combined and compared, generating new seminorms and producing equivalence relations at the level of topology.

9.1 Sum, maximum, and scaling of seminorms

If \(p\) and \(q\) are seminorms and \(c\ge 0\), then:

  • \(p+q\) is a seminorm,
  • \(\max(p,q)\) is a seminorm,
  • \(cp\) is a seminorm.

These operations preserve subadditivity and homogeneity while affecting the kernel in predictable ways.

9.2 Pointwise infimum/supremum constructions (where applicable)

Under suitable conditions, taking pointwise infima or suprema across families of seminorms can produce another seminorm, or at least a function with controlled subadditivity. In many topological applications, however, it is the induced family (rather than a single collapsed formula) that is used to describe the topology accurately.

9.3 Lattice-style properties and domination relations

Seminorms form an order structure under pointwise comparison: \(p\le q\) means \(p(x)\le q(x)\) for all \(x\). This order translates into inclusion relations between sublevel sets and can be used to express domination of one seminorm by another. Lattice operations like \(\max\) align with these order-theoretic ideas.

9.4 “Equivalent” seminorms in the sense of topological equivalence

Two seminorms (or families) may be considered equivalent when they yield the same convergence behavior and neighborhood structure near zero. For a pair of seminorms \(p\) and \(q\) this can be expressed by inequalities each controlling the other up to constants, while in more general locally convex settings it involves matching the generated topologies rather than matching kernels alone.

10 Applications and typical roles

Seminorms serve as a flexible foundation for metric-like structures, continuity notions, and the translation of algebraic constraints into topological behavior.

10.1 Defining metrics from seminorms

Even when a seminorm is degenerate, it yields a pseudo-metric on the original space. After quotienting by the kernel, the pseudo-metric becomes a genuine metric associated with a norm on the factor space. This method is widely used to turn “relative size” measurements into true distance functions.

10.2 Building Fréchet and other locally convex spaces (high level)

Many classical function spaces—especially those arising from smoothness and convergence requirements—are modeled as locally convex spaces generated by countable seminorm families. When the family is countable and the topology is complete in an appropriate sense, one obtains Fréchet spaces, which are central in analysis and partial differential equations.

10.3 Characterizing continuity and convergence in function spaces

Seminorms provide concrete criteria for convergence in spaces of functions, distributions, or operators: a sequence converges when all relevant seminorms tend to the required limits. Continuity of maps is likewise expressed through seminorm estimates, making proofs modular and robust.

10.4 Bridging between algebraic constraints and topological behavior

Because seminorm kernels correspond to algebraic subspaces, seminorm-based topology naturally reflects which directions are “identified” or “ignored” by the measurement. Quotient constructions translate algebraic degeneracy into topological Hausdorffness, while families of seminorms allow separate constraints to be combined into a coherent notion of smallness.