trait
NormedVectorSpace[V, F] extends VectorSpace[V, F] with MetricSpace[V, F]
Abstract Value Members
-
abstract
def
negate(x: V): V
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abstract
def
norm(v: V): F
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abstract
def
plus(x: V, y: V): V
-
implicit abstract
def
scalar: Field[F]
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abstract
def
timesl(r: F, v: V): V
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abstract
def
zero: V
Concrete Value Members
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final
def
!=(arg0: AnyRef): Boolean
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final
def
!=(arg0: Any): Boolean
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final
def
##(): Int
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final
def
==(arg0: AnyRef): Boolean
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final
def
==(arg0: Any): Boolean
-
def
additive: AbGroup[V]
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final
def
asInstanceOf[T0]: T0
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def
clone(): AnyRef
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def
distance(v: V, w: V): F
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def
divr(v: V, f: F): V
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final
def
eq(arg0: AnyRef): Boolean
-
def
equals(arg0: Any): Boolean
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def
finalize(): Unit
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final
def
getClass(): Class[_]
-
def
hashCode(): Int
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final
def
isInstanceOf[T0]: Boolean
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def
minus(x: V, y: V): V
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final
def
ne(arg0: AnyRef): Boolean
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def
normalize(v: V): V
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final
def
notify(): Unit
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final
def
notifyAll(): Unit
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final
def
synchronized[T0](arg0: ⇒ T0): T0
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def
timesr(v: V, r: F): V
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def
toString(): String
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final
def
wait(): Unit
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final
def
wait(arg0: Long, arg1: Int): Unit
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final
def
wait(arg0: Long): Unit
Inherited from AnyRef
Inherited from Any
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 isnorm(k *: v) == k.abs * norm(v)
. Additionally, a normed vector space is also aMetricSpace
, wheredistance(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.