trait NormedVectorSpace[V, F] extends VectorSpace[V, F] with MetricSpace[V, F]

A normed vector space is a vector space equipped with a function norm: V => F. The main constraint is that the norm of a vector must be scaled linearly when the vector is scaled; that is norm(k *: v) == k.abs * norm(v). Additionally, a normed vector space is also a MetricSpace, where distance(v, w) = norm(v - w), and so must obey the triangle inequality.

An example of a normed vector space is R^n equipped with the euclidean vector length as the norm.

Linear Supertypes
MetricSpace[V, F], VectorSpace[V, F], Module[V, F], AdditiveAbGroup[V], AdditiveCMonoid[V], AdditiveCSemigroup[V], AdditiveGroup[V], AdditiveMonoid[V], AdditiveSemigroup[V], Any
Known Subclasses
DistNormedVectorSpace, SeqLpNormedVectorSpace, SeqMaxNormedVectorSpace
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. NormedVectorSpace
  2. MetricSpace
  3. VectorSpace
  4. Module
  5. AdditiveAbGroup
  6. AdditiveCMonoid
  7. AdditiveCSemigroup
  8. AdditiveGroup
  9. AdditiveMonoid
  10. AdditiveSemigroup
  11. Any
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Visibility
  1. Public
  2. All

Abstract Value Members

  1. abstract def getClass(): Class[_]
    Definition Classes
    Any
  2. abstract def negate(x: V): V
    Definition Classes
    AdditiveGroup
  3. abstract def norm(v: V): F
  4. abstract def plus(x: V, y: V): V
    Definition Classes
    AdditiveSemigroup
  5. implicit abstract def scalar: Field[F]
    Definition Classes
    VectorSpace → Module
  6. abstract def timesl(r: F, v: V): V
    Definition Classes
    Module
  7. abstract def zero: V
    Definition Classes
    AdditiveMonoid

Concrete Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    Any
  2. final def ##(): Int
    Definition Classes
    Any
  3. final def ==(arg0: Any): Boolean
    Definition Classes
    Any
  4. def additive: AbGroup[V]
    Definition Classes
    AdditiveAbGroup → AdditiveCMonoid → AdditiveCSemigroup → AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
  5. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  6. def distance(v: V, w: V): F
    Definition Classes
    NormedVectorSpace → MetricSpace
  7. def divr(v: V, f: F): V
    Definition Classes
    VectorSpace
  8. def equals(arg0: Any): Boolean
    Definition Classes
    Any
  9. def hashCode(): Int
    Definition Classes
    Any
  10. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  11. def isZero(a: V)(implicit ev: Eq[V]): Boolean

    Tests if a is zero.

    Tests if a is zero.

    Definition Classes
    AdditiveMonoid
  12. def minus(x: V, y: V): V
    Definition Classes
    AdditiveGroup
  13. def normalize(v: V): V
  14. def sum(as: TraversableOnce[V]): V

    Given a sequence of as, sum them using the monoid and return the total.

    Given a sequence of as, sum them using the monoid and return the total.

    Definition Classes
    AdditiveMonoid
  15. def sumOption(as: TraversableOnce[V]): Option[V]

    Given a sequence of as, sum them using the semigroup and return the total.

    Given a sequence of as, sum them using the semigroup and return the total.

    If the sequence is empty, returns None. Otherwise, returns Some(total).

    Definition Classes
    AdditiveSemigroup
  16. def sumn(a: V, n: Int): V

    Return a added with itself n times.

    Return a added with itself n times.

    Definition Classes
    AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
  17. def sumnAboveOne(a: V, n: Int): V
    Attributes
    protected
    Definition Classes
    AdditiveSemigroup
  18. def timesr(v: V, r: F): V
    Definition Classes
    Module
  19. def toString(): String
    Definition Classes
    Any

Inherited from MetricSpace[V, F]

Inherited from VectorSpace[V, F]

Inherited from Module[V, F]

Inherited from AdditiveAbGroup[V]

Inherited from AdditiveCMonoid[V]

Inherited from AdditiveCSemigroup[V]

Inherited from AdditiveGroup[V]

Inherited from AdditiveMonoid[V]

Inherited from AdditiveSemigroup[V]

Inherited from Any

Ungrouped