spire.algebra

NormedVectorSpace

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], AdditiveCommutativeGroup[V], AdditiveCommutativeMonoid[V], AdditiveCommutativeSemigroup[V], algebra.ring.AdditiveGroup[V], algebra.ring.AdditiveMonoid[V], algebra.ring.AdditiveSemigroup[V], Serializable, Serializable, Any
Known Subclasses
DistNormedVectorSpace, SeqLpNormedVectorSpace, SeqMaxNormedVectorSpace
Ordering
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Inherited
  1. NormedVectorSpace
  2. MetricSpace
  3. VectorSpace
  4. Module
  5. AdditiveCommutativeGroup
  6. AdditiveCommutativeMonoid
  7. AdditiveCommutativeSemigroup
  8. AdditiveGroup
  9. AdditiveMonoid
  10. AdditiveSemigroup
  11. Serializable
  12. Serializable
  13. Any
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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: CommutativeGroup[V]

    Definition Classes
    AdditiveCommutativeGroup → AdditiveCommutativeMonoid → AdditiveCommutativeSemigroup → 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: algebra.Eq[V]): Boolean

    Definition Classes
    AdditiveMonoid

  12. def minus(x: V, y: V): V

    Definition Classes
    AdditiveGroup

  13. def normalize(v: V): V

  14. def positiveSumN(a: V, n: Int): V

    Attributes
    protected[this]
    Definition Classes
    AdditiveSemigroup

  15. def sum(as: TraversableOnce[V]): V

    Definition Classes
    AdditiveMonoid

  16. def sumN(a: V, n: Int): V

    Definition Classes
    AdditiveGroup → AdditiveMonoid → AdditiveSemigroup

  17. def timesr(v: V, r: F): V

    Definition Classes
    Module

  18. def toString(): String

    Definition Classes
    Any

  19. def trySum(as: TraversableOnce[V]): Option[V]

    Definition Classes
    AdditiveMonoid → AdditiveSemigroup

Inherited from MetricSpace[V, F]

Inherited from VectorSpace[V, F]

Inherited from Module[V, F]

Inherited from AdditiveCommutativeGroup[V]

Inherited from AdditiveCommutativeMonoid[V]

Inherited from AdditiveCommutativeSemigroup[V]

Inherited from algebra.ring.AdditiveGroup[V]

Inherited from algebra.ring.AdditiveMonoid[V]

Inherited from algebra.ring.AdditiveSemigroup[V]

Inherited from Serializable

Inherited from Serializable

Inherited from Any

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