NormedVectorSpace

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

Abstract Value Members

abstract def getClass(): Class[_]

Definition Classes
Any
abstract def negate(x: V): V

Definition Classes
AdditiveGroup
abstract def norm(v: V): F
abstract def plus(x: V, y: V): V

Definition Classes
AdditiveSemigroup
implicit abstract def scalar: Field[F]

Definition Classes
VectorSpace → Module
abstract def timesl(r: F, v: V): V

Definition Classes
Module
abstract def zero: V

Definition Classes
AdditiveMonoid

Concrete Value Members

final def !=(arg0: Any): Boolean

Definition Classes
Any
final def ##(): Int

Definition Classes
Any
final def ==(arg0: Any): Boolean

Definition Classes
Any
def additive: AbGroup[V]

Definition Classes
AdditiveAbGroup → AdditiveCMonoid → AdditiveCSemigroup → AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
final def asInstanceOf[T0]: T0

Definition Classes
Any
def distance(v: V, w: V): F

Definition Classes
NormedVectorSpace → MetricSpace
def divr(v: V, f: F): V

Definition Classes
VectorSpace
def equals(arg0: Any): Boolean

Definition Classes
Any
def hashCode(): Int

Definition Classes
Any
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isZero(a: V)(implicit ev: Eq[V]): Boolean

Tests if a is zero.
Tests if a is zero.

Definition Classes
AdditiveMonoid
def minus(x: V, y: V): V

Definition Classes
AdditiveGroup
def normalize(v: V): V
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
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
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
def sumnAboveOne(a: V, n: Int): V

Attributes
protected
Definition Classes
AdditiveSemigroup
def timesr(v: V, r: F): V

Definition Classes
Module
def toString(): String

Definition Classes
Any

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

Abstract Value Members

abstract def getClass(): Class[_]

abstract def negate(x: V): V

abstract def norm(v: V): F

abstract def plus(x: V, y: V): V

implicit abstract def scalar: Field[F]

abstract def timesl(r: F, v: V): V

abstract def zero: V

Concrete Value Members

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

def additive: AbGroup[V]

final def asInstanceOf[T0]: T0

def distance(v: V, w: V): F

def divr(v: V, f: F): V

def equals(arg0: Any): Boolean

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

def isZero(a: V)(implicit ev: Eq[V]): Boolean

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

def normalize(v: V): V

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

def sumOption(as: TraversableOnce[V]): Option[V]

def sumn(a: V, n: Int): V

def sumnAboveOne(a: V, n: Int): V

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

def toString(): String

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