Minkowski functional

Short description: Function made from a set

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In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.

If K is a subset of a real or complex vector space X, then the Minkowski functional or gauge of K is defined to be the function pK:X->[0,∞], valued in the extended real numbers, defined by pK(x)=inf⁡{r∈ℝ:r>0 and x∈rK},x∈X, where the infimum of the empty set is defined to be positive infinity.

The set K is often assumed to have properties, such as being an absorbing disk in X, that guarantee that pK will be a seminorm on X. In fact, every seminorm p on X is equal to the Minkowski functional (that is, p=pK) of any subset K of X satisfying

{x∈X:p(x)<1}⊆K⊆{x∈X:p(x)≤1}

(where all three of these sets are necessarily absorbing in X and the first and last are also disks).

Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of X into certain algebraic properties of a function on X.

The Minkowski function is always non-negative (meaning pK≥0). This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values. However, pK might not be real-valued since for any given x∈X, the value pK(x) is a real number if and only if {r>0:x∈rK} is not empty. Consequently, K is usually assumed to have properties (such as being absorbing in X, for instance) that will guarantee that pK is real-valued.

## Contents

* 1 Definition
* 1.1 Some conditions making a gauge real-valued
* 2 Motivating examples
* 2.1 Example 1
* 2.2 Example 2
* 3 Common conditions guaranteeing gauges are seminorms
* 3.1 Gauges of absorbing disks
* 3.2 Algebraic properties
* 3.3 Topological properties
* 4 Minimal requirements on the set
* 4.1 Properties
* 4.2 Examples
* 4.3 Positive homogeneity characterizes Minkowski functionals
* 4.4 Characterizing Minkowski functionals on star sets
* 4.5 Characterizing Minkowski functionals that are seminorms
* 4.6 Positive sublinear functions and Minkowski functionals
* 4.7 Correspondence between open convex sets and positive continuous sublinear functions
* 5 See also
* 6 Notes
* 7 References
* 8 Further reading

## Definition

Let K be a subset of a real or complex vector space X. Define the gauge of K or the Minkowski functional associated with or induced by K as being the function pK:X->[0,∞], valued in the extended real numbers, defined by

pK(x):=inf⁡{r>0:x∈rK},

(recall that the infimum of the empty set is ∞, that is, inf⁡∅=∞). Here, {r>0:x∈rK} is shorthand for {r∈ℝ:r>0 and x∈rK}.

For any x∈X, pK(x)≠∞ if and only if {r>0:x∈rK} is not empty. The arithmetic operations on ℝ can be extended to operate on ±∞, where r±∞:=0 for all non-zero real −∞<r<∞. The products 0⋅∞ and 0⋅−∞ remain undefined.

### Some conditions making a gauge real-valued

In the field of convex analysis, the map pK taking on the value of ∞ is not necessarily an issue. However, in functional analysis pK is almost always real-valued (that is, to never take on the value of ∞), which happens if and only if the set {r>0:x∈rK} is non-empty for every x∈X.

In order for pK to be real-valued, it suffices for the origin of X to belong to the algebraic interior or core of K in X.[1] If K is absorbing in X, where recall that this implies that 0∈K, then the origin belongs to the algebraic interior of K in X and thus pK is real-valued. Characterizations of when pK is real-valued are given below.

## Motivating examples

### Example 1

Consider a normed vector space (X,‖⋅‖), with the norm ‖⋅‖ and let U:={x∈X:‖x‖≤1} be the unit ball in X. Then for every x∈X, ‖x‖=pU(x). Thus the Minkowski functional pU is just the norm on X.

### Example 2

Let X be a vector space without topology with underlying scalar field 𝕂. Let f:X->𝕂 be any linear functional on X (not necessarily continuous). Fix a>0. Let K be the set K:={x∈X:|f(x)|≤a} and let pK be the Minkowski functional of K. Then pK(x)=1a|f(x)| for all x∈X. The function pK has the following properties:

1. It is subadditive: pK(x+y)≤pK(x)+pK(y).
2. It is absolutely homogeneous: pK(sx)=|s|pK(x) for all scalars s.
3. It is nonnegative: pK≥0.

Therefore, pK is a seminorm on X, with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets. What is meant precisely by "nice" is discussed in the section below.

Notice that, in contrast to a stronger requirement for a norm, pK(x)=0 need not imply x=0. In the above example, one can take a nonzero x from the kernel of f. Consequently, the resulting topology need not be Hausdorff.

## Common conditions guaranteeing gauges are seminorms

To guarantee that pK(0)=0, it will henceforth be assumed that 0∈K.

In order for pK to be a seminorm, it suffices for K to be a disk (that is, convex and balanced) and absorbing in X, which are the most common assumption placed on K.

Theorem[2] -- If K is an absorbing disk in a vector space X then the Minkowski functional of K, which is the map pK:X->[0,∞) defined by pK(x):=inf⁡{r>0:x∈rK}, is a seminorm on X. Moreover, pK(x)=1sup⁡{r>0:rx∈K}.

More generally, if K is convex and the origin belongs to the algebraic interior of K, then pK is a nonnegative sublinear functional on X, which implies in particular that it is subadditive and positive homogeneous. If K is absorbing in X then p[0,1]K is positive homogeneous, meaning that p[0,1]K(sx)=sp[0,1]K(x) for all real s≥0, where [0,1]K={tk:t∈[0,1],k∈K}.[3] If q is a nonnegative real-valued function on X that is positive homogeneous, then the sets U:={x∈X:q(x)<1} and D:={x∈X:q(x)≤1} satisfy [0,1]U=U and [0,1]D=D; if in addition q is absolutely homogeneous then both U and D are balanced.[3]

### Gauges of absorbing disks

Arguably the most common requirements placed on a set K to guarantee that pK is a seminorm are that K be an absorbing disk in X. Due to how common these assumptions are, the properties of a Minkowski functional pK when K is an absorbing disk will now be investigated. Since all of the results mentioned above made few (if any) assumptions on K, they can be applied in this special case.

Theorem -- Assume that K is an absorbing subset of X. It is shown that:

1. If K is convex then pK is subadditive.
2. If K is balanced then pK is absolutely homogeneous; that is, pK(sx)=|s|pK(x) for all scalars s.

Proof that the Gauge of an absorbing disk is a seminorm

Convexity and subadditivity

A simple geometric argument that shows convexity of K implies subadditivity is as follows. Suppose for the moment that pK(x)=pK(y)=r. Then for all e>0, x,y∈Ke:=(r,e)K. Since K is convex and r+e≠0, Ke is also convex. Therefore, 12x+12y∈Ke. By definition of the Minkowski functional pK, pK(12x+12y)≤r+e=12pK(x)+12pK(y)+e.

But the left hand side is 12pK(x+y), so that pK(x+y)≤pK(x)+pK(y)+2e.

Since e>0 was arbitrary, it follows that pK(x+y)≤pK(x)+pK(y), which is the desired inequality. The general case pK(x)>pK(y) is obtained after the obvious modification.

Convexity of K, together with the initial assumption that the set {r>0:x∈rK} is nonempty, implies that K is absorbing.

Balancedness and absolute homogeneity

Notice that K being balanced implies that λx∈rKif and only ifx∈r|λ|K.

Therefore pK(λx)=inf⁡{r>0:λx∈rK}=inf⁡{r>0:x∈r|λ|K}=inf⁡{|λ|r|λ|>0:x∈r|λ|K}=|λ|pK(x).

### Algebraic properties

Let X be a real or complex vector space and let K be an absorbing disk in X.

* pK is a seminorm on X.
* pK is a norm on X if and only if K does not contain a non-trivial vector subspace.[4]
* psK=1|s|pK for any scalar s≠0.[4]
* If J is an absorbing disk in X and J⊆K then pK≤pJ.
* If K is a set satisfying {x∈X:p(x)<1}⊆K⊆{x∈X:p(x)≤1} then K is absorbing in X and p=pK, where pK is the Minkowski functional associated with K; that is, it is the gauge of K.[5]
* In particular, if K is as above and q is any seminorm on X, then q=p if and only if {x∈X:q(x)<1}⊆K⊆{x∈X:q(x)≤1}.[5]
* If x∈X satisfies pK(x)<1 then x∈K.

### Topological properties

Assume that X is a (real or complex) topological vector space (not necessarily Hausdorff or locally convex) and let K be an absorbing disk in X. Then

IntXK⊆{x∈X:pK(x)<1}⊆K⊆{x∈X:pK(x)≤1}⊆ClXK,

where IntXK is the topological interior and ClXK is the topological closure of K in X.[6] Importantly, it was not assumed that pK was continuous nor was it assumed that K had any topological properties.

Moreover, the Minkowski functional pK is continuous if and only if K is a neighborhood of the origin in X.[6] If pK is continuous then[6] IntXK={x∈X:pK(x)<1} and ClXK={x∈X:pK(x)≤1}.

## Minimal requirements on the set

This section will investigate the most general case of the gauge of any subset K of X. The more common special case where K is assumed to be an absorbing disk in X was discussed above.

### Properties

All results in this section may be applied to the case where K is an absorbing disk.

Throughout, K is any subset of X.

Summary -- Suppose that K is a subset of a real or complex vector space X.

1. Strict positive homogeneity: pK(rx)=rpK(x) for all x∈X and all positive real r>0.
* Positive/Nonnegative homogeneity: pK is nonnegative homogeneous if and only if pK is real-valued.
* A map p is called nonnegative homogeneous[7] if p(rx)=rp(x) for all x∈X and all nonnegative real r≥0. Since 0⋅∞ is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
2. Real-values: (0,∞)K is the set of all points on which pK is real valued. So pK is real-valued if and only if (0,∞)K=X, in which case 0∈K.
* Value at 0: pK(0)≠∞ if and only if 0∈K if and only if pK(0)=0.
* Null space: If x∈X then pK(x)=0 if and only if (0,∞)x⊆(0,1)K if and only if there exists a divergent sequence of positive real numbers t1,t2,t3,⋯->∞ such that tnx∈K for all n. Moreover, the zero set of pK is ker⁡pK=def{y∈X:pK(y)=0}=⋂e>0(0,e)K.
3. Comparison to a constant: If 0≤r≤∞ then for any x∈X, pK(x)<r if and only if x∈(0,r)K; this can be restated as: If 0≤r≤∞ then pK−1([0,r))=(0,r)K.
* It follows that if 0≤R<∞ is real then pK−1([0,R])=⋂e>0(0,R+e)K, where the set on the right hand side denotes ⋂e>0[(0,R+e)K] and not its subset [⋂e>0(0,R+e)]K=(0,R]K. If R>0 then these sets are equal if and only if K contains {y∈X:pK(y)=1}.
* In particular, if x∈RK or x∈(0,R]K then pK(x)≤R, but importantly, the converse is not necessarily true.
4. Gauge comparison: For any subset L⊆X, pK≤pL if and only if (0,1)L⊆(0,1)K; thus pL=pK if and only if (0,1)L=(0,1)K.
* The assignment L↦pL is order-reversing in the sense that if K⊆L then pL≤pK.[8]
* Because the set L:=(0,1)K satisfies (0,1)L=(0,1)K, it follows that replacing K with pK−1([0,1))=(0,1)K will not change the resulting Minkowski functional. The same is true of L:=(0,1]K and of L:=pK−1([0,1]).
* If D=def{y∈X:pK(y)=1 or pK(y)=0} then pD=pK and D has the particularly nice property that if r>0 is real then x∈rD if and only if pD(x)=r or pD(x)=0.[note 1] Moreover, if r>0 is real then pD(x)≤r if and only if x∈(0,r]D.
5. Subadditive/Triangle inequality: pK is subadditive if and only if (0,1)K is convex. If K is convex then so are both (0,1)K and (0,1]K and moreover, pK is subadditive.
6. Scaling the set: If s≠0 is a scalar then psK(y)=pK(1sy) for all y∈X. Thus if 0<r<∞ is real then prK(y)=pK(1ry)=1rpK(y).
7. Symmetric: pK is symmetric (meaning that pK(−y)=pK(y) for all y∈X) if and only if (0,1)K is a symmetric set (meaning that(0,1)K=−(0,1)K), which happens if and only if pK=p−K.
8. Absolute homogeneity: pK(ux)=pK(x) for all x∈X and all unit length scalars u[note 2] if and only if (0,1)uK⊆(0,1)K for all unit length scalars u, in which case pK(sx)=|s|pK(x) for all x∈X and all non-zero scalars s≠0. If in addition pK is also real-valued then this holds for all scalars s (that is, pK is absolutely homogeneous[note 3]).
* (0,1)uK⊆(0,1)K for all unit length u if and only if (0,1)uK=(0,1)K for all unit length u.
* sK⊆K for all unit scalars s if and only if sK=K for all unit scalars s; if this is the case then (0,1)K=(0,1)sK for all unit scalars s.
* The Minkowski functional of any balanced set is a balanced function.[8]
9. Absorbing: If K is convex or balanced and if (0,∞)K=X then K is absorbing in X.
* If a set A is absorbing in X and A⊆K then K is absorbing in X.
* If K is convex and 0∈K then [0,1]K=K, in which case (0,1)K⊆K.
10. Restriction to a vector subspace: If S is a vector subspace of X and if pK∩S:S->[0,∞] denotes the Minkowski functional of K∩S on S, then pK|S=pK∩S, where pK|S denotes the restriction of pK to S.

Proof

The proofs of these basic properties are straightforward exercises so only the proofs of the most important statements are given.

The proof that a convex subset A⊆X that satisfies (0,∞)A=X is necessarily absorbing in X is straightforward and can be found in the article on absorbing sets.

For any real t>0,

{r>0:tx∈rK}={t(r/t):x∈(r/t)K}=t{s>0:x∈sK}

so that taking the infimum of both sides shows that

pK(tx)=inf⁡{r>0:tx∈rK}=tinf⁡{s>0:x∈sK}=tpK(x).

This proves that Minkowski functionals are strictly positive homogeneous. For 0⋅pK(x) to be well-defined, it is necessary and sufficient that pK(x)≠∞; thus pK(tx)=tpK(x) for all x∈X and all non-negative real t≥0 if and only if pK is real-valued.

The hypothesis of statement (7) allows us to conclude that pK(sx)=pK(x) for all x∈X and all scalars s satisfying |s|=1. Every scalar s is of the form reit for some real t where r:=|s|≥0 and eit is real if and only if s is real. The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of pK, and from the positive homogeneity of pK when pK is real-valued. ◼

### Examples

1. If ℒ is a non-empty collection of subsets of X then p∪ℒ(x)=inf⁡{pL(x):L∈ℒ} for all x∈X, where ∪ℒ=def⋃L∈ℒL.
* Thus pK∪L(x)=min⁡{pK(x),pL(x)} for all x∈X.
2. If ℒ is a non-empty collection of subsets of X and I⊆X satisfies

{x∈X:pL(x)<1 for all L∈ℒ}⊆I⊆{x∈X:pL(x)≤1 for all L∈ℒ} then pI(x)=sup⁡{pL(x):L∈ℒ} for all x∈X.

The following examples show that the containment (0,R]K⊆⋂e>0(0,R+e)K could be proper.

Example: If R=0 and K=X then (0,R]K=(0,0]X=∅X=∅ but ⋂e>0(0,e)K=⋂e>0X=X, which shows that its possible for (0,R]K to be a proper subset of ⋂e>0(0,R+e)K when R=0. ◼

The next example shows that the containment can be proper when R=1; the example may be generalized to any real R>0. Assuming that [0,1]K⊆K, the following example is representative of how it happens that x∈X satisfies pK(x)=1 but x∉(0,1]K.

Example: Let x∈X be non-zero and let K=[0,1)x so that [0,1]K=K and x∉K. From x∉(0,1)K=K it follows that pK(x)≥1. That pK(x)≤1 follows from observing that for every e>0, (0,1+e)K=[0,1+e)([0,1)x)=[0,1+e)x, which contains x. Thus pK(x)=1 and x∈⋂e>0(0,1+e)K. However, (0,1]K=(0,1]([0,1)x)=[0,1)x=K so that x∉(0,1]K, as desired. ◼

### Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are exactly those functions f:X->[0,∞] that have a certain purely algebraic property that is commonly encountered.

Theorem -- Let f:X->[0,∞] be any function. The following statements are equivalent:

1. Strict positive homogeneity: f(tx)=tf(x) for all x∈X and all positive real t>0.
* This statement is equivalent to: f(tx)≤tf(x) for all x∈X and all positive real t>0.
2. f is a Minkowski functional: meaning that there exists a subset S⊆X such that f=pS.
3. f=pK where K:={x∈X:f(x)≤1}.
4. f=pV where V:={x∈X:f(x)<1}.

Moreover, if f never takes on the value ∞ (so that the product 0⋅f(x) is always well-defined) then this list may be extended to include:

5. Positive/Nonnegative homogeneity: f(tx)=tf(x) for all x∈X and all nonnegative real t≥0.

Proof

If f(tx)≤tf(x) holds for all x∈X and real t>0 then tf(x)=tf(1t(tx))≤t1tf(tx)=f(tx)≤tf(x) so that tf(x)=f(tx).

Only (1) implies (3) will be proven because afterwards, the rest of the theorem follows immediately from the basic properties of Minkowski functionals described earlier; properties that will henceforth be used without comment. So assume that f:X->[0,∞] is a function such that f(tx)=tf(x) for all x∈X and all real t>0 and let K:={y∈X:f(y)≤1}.

For all real t>0, f(0)=f(t0)=tf(0) so by taking t=2 for instance, it follows that either f(0)=0 or f(0)=∞. Let x∈X. It remains to show that f(x)=pK(x).

It will now be shown that if f(x)=0 or f(x)=∞ then f(x)=pK(x), so that in particular, it will follow that f(0)=pK(0). So suppose that f(x)=0 or f(x)=∞; in either case f(tx)=tf(x)=f(x) for all real t>0. Now if f(x)=0 then this implies that that tx∈K for all real t>0 (since f(tx)=0≤1), which implies that pK(x)=0, as desired. Similarly, if f(x)=∞ then tx∉K for all real t>0, which implies that pK(x)=∞, as desired. Thus, it will henceforth be assumed that R:=f(x) a positive real number and that x≠0 (importantly, however, the possibility that pK(x) is 0 or ∞ has not yet been ruled out).

Recall that just like f, the function pK satisfies pK(tx)=tpK(x) for all real t>0. Since 0<1R<∞, pK(x)=R=f(x) if and only if pK(1Rx)=1=f(1Rx) so assume without loss of generality that R=1 and it remains to show that pK(1Rx)=1. Since f(x)=1, x∈K⊆(0,1]K, which implies that pK(x)≤1 (so in particular, pK(x)≠∞ is guaranteed). It remains to show that pK(x)≥1, which recall happens if and only if x∉(0,1)K. So assume for the sake of contradiction that x∈(0,1)K and let 0<r<1 and k∈K be such that x=rk, where note that k∈K implies that f(k)≤1. Then 1=f(x)=f(rk)=rf(k)≤r<1. ◼

This theorem can be extended to characterize certain classes of [−∞,∞]-valued maps (for example, real-valued sublinear functions) in terms of Minkowski functionals. For instance, it can be used to describe how every real homogeneous function f:X->ℝ (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

### Characterizing Minkowski functionals on star sets

Proposition[10] -- Let f:X->[0,∞] be any function and K⊆X be any subset. The following statements are equivalent:

1. f is (strictly) positive homogeneous, f(0)=0, and

{x∈X:f(x)<1}⊆K⊆{x∈X:f(x)≤1}.

2. f is the Minkowski functional of K (that is, f=pK), K contains the origin, and K is star-shaped at the origin.
* The set K is star-shaped at the origin if and only if tk∈K whenever k∈K and 0≤t≤1. A set that is star-shaped at the origin is sometimes called a star set.[9]

### Characterizing Minkowski functionals that are seminorms

In this next theorem, which follows immediately from the statements above, K is not assumed to be absorbing in X and instead, it is deduced that (0,1)K is absorbing when pK is a seminorm. It is also not assumed that K is balanced (which is a property that K is often required to have); in its place is the weaker condition that (0,1)sK⊆(0,1)K for all scalars s satisfying |s|=1. The common requirement that K be convex is also weakened to only requiring that (0,1)K be convex.

Theorem -- Let K be a subset of a real or complex vector space X. Then pK is a seminorm on X if and only if all of the following conditions hold:

1. (0,∞)K=X (or equivalently, pK is real-valued).
2. (0,1)K is convex (or equivalently, pK is subadditive).
* It suffices (but is not necessary) for K to be convex.
3. (0,1)uK⊆(0,1)K for all unit scalars u.
* This condition is satisfied if K is balanced or more generally if uK⊆K for all unit scalars u.

in which case 0∈K and both (0,1)K={x∈X:p(x)<1} and ⋂e>0(0,1+e)K={x∈X:pK(x)≤1} will be convex, balanced, and absorbing subsets of X.

Conversely, if f is a seminorm on X then the set V:={x∈X:f(x)<1} satisfies all three of the above conditions (and thus also the conclusions) and also f=pV; moreover, V is necessarily convex, balanced, absorbing, and satisfies (0,1)V=V=[0,1]V.

Corollary -- If K is a convex, balanced, and absorbing subset of a real or complex vector space X, then pK is a seminorm on X.

### Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function f:X->ℝ on an arbitrary topological vector space X is continuous at the origin if and only if it is uniformly continuous, where if in addition f is nonnegative, then f is continuous if and only if V:={x∈X:f(x)<1} is an open neighborhood in X.[11] If f:X->ℝ is subadditive and satisfies f(0)=0, then f is continuous if and only if its absolute value |f|:X->[0,∞) is continuous.

A nonnegative sublinear function is a nonnegative homogeneous function f:X->[0,∞) that satisfies the triangle inequality. It follows immediately from the results below that for such a function f, if V:={x∈X:f(x)<1} then f=pV. Given K⊆X, the Minkowski functional pK is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if (0,∞)K=X and (0,1)K is convex.

### Correspondence between open convex sets and positive continuous sublinear functions

Theorem[11] -- Suppose that X is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the non-empty open convex subsets of X are exactly those sets that are of the form z+{x∈X:p(x)<1}={x∈X:p(x−z)<1} for some z∈X and some positive continuous sublinear function p on X.

Proof

Let V≠∅ be an open convex subset of X. If 0∈V then let z:=0 and otherwise let z∈V be arbitrary. Let p=pK:X->[0,∞) be the Minkowski functional of K:=V−z where this convex open neighborhood of the origin satisfies (0,1)K=K. Then p is a continuous sublinear function on X since V−z is convex, absorbing, and open (however, p is not necessarily a seminorm since it is not necessarily absolutely homogeneous). From the properties of Minkowski functionals, we have pK−1([0,1))=(0,1)K, from which it follows that V−z={x∈X:p(x)<1} and so V=z+{x∈X:p(x)<1}. Since z+{x∈X:p(x)<1}={x∈X:p(x−z)<1}, this completes the proof. ◼

## See also

* Asymmetric norm - Generalization of the concept of a norm
* Auxiliary normed space
* Cauchy's functional equation - Functional equation
* Finsler manifold - Generalization of Riemannian manifolds
* Hadwiger's theorem
* Biography:Hugo Hadwiger - Swiss mathematician (1908–1981)
* Locally convex topological vector space - Vector space with a topology defined by convex open sets
* Norm (mathematics) - Length in a vector space
* Seminorm
* Topological vector space - Vector space with a notion of nearness

## Notes

1. ↑ It is in general false that x∈rD if and only if pD(x)=r (for example, consider when pK is a norm or a seminorm). The correct statement is: If 0<r<∞ then x∈rD if and only if pD(x)=r or pD(x)=0.
2. ↑ u is having unit length means that |u|=1.
3. ↑ The map pK is called absolutely homogeneous if |s|pK(x) is well-defined and pK(sx)=|s|pK(x) for all x∈X and all scalars s (not just non-zero scalars).

## References

1. ↑ Narici & Beckenstein 2011, p. 109.
2. ↑ Narici & Beckenstein 2011, p. 119.
3. ↑ 3.0 3.1 Jarchow 1981, pp. 104–108.
4. ↑ 4.0 4.1 Narici & Beckenstein 2011, pp. 115–154.
5. ↑ 5.0 5.1 Schaefer 1999, p. 40.
6. ↑ 6.0 6.1 6.2 Narici & Beckenstein 2011, p. 119-120.
7. ↑ Kubrusly 2011, p. 200.
8. ↑ 8.0 8.1 Schechter 1996, p. 316.
9. ↑ Schechter 1996, p. 303.
10. ↑ Schechter 1996, pp. 313–317.
11. ↑ 11.0 11.1 Narici & Beckenstein 2011, pp. 192–193.

* Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
* Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0.
* Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
* Template:Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited
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* Template:Dunford Schwartz Linear Operators Part 1 General Theory
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* Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. https://archive.org/details/topologicalvecto0000grot.
* Template:Hogbe-Nlend Bornologies and Functional Analysis
* Template:Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces
* Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
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* Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
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* Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
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## Further reading

* F. Simeski, A. M. P. Boelens, and M. Ihme. "Modeling Adsorption in Silica Pores via Minkowski Functionals and Molecular Electrostatic Moments". Energies 13 (22) 5976 (2020). doi:10.3390/en13225976.

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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
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* (Dis)Continuous
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* Normal
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* Self-adjoint
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* Transpose
* Unitary

Operator theory|

* Banach algebras
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* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
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* Bessel's inequality
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* Closed graph
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Analysis|

* Abstract Wiener space
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* holomorphic)
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* Dunford
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* weak)
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* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
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* Vector)
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Types of sets|

* Absolutely convex
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* Bounded
* Convex
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* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
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* Extreme point
* Interior
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* Polar

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