Numerical Methods in Computational Finance - Daniel J. Duffy

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V left-parenthesis upper K right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column upper A 2 colon x plus y equals y plus x 3rd Row 1st Column Blank 2nd Column upper A 3 colon Exists unique 0 in upper V such that 0 plus x equals x plus 0 equals x 4th Row 1st Column Blank 2nd Column upper A 4 colon For each x in upper V there exists a unique y such that x plus y equals 0 5th Row 1st Column Blank 2nd Column left-parenthesis italic the negative of x right-parenthesis comma called negative x period EndLayout"/>

      Axiom A1 states that addition is associative, and axiom A2 states that addition is commutative. The element 0 is called the zero element of the vector space.

      Scalar multiplication is defined by the axioms left-parenthesis a comma b element-of upper K and x comma y element-of upper V right-parenthesis:

      (4.3)StartLayout 1st Row 1st Column Blank 2nd Column normal upper B Baseline 1 colon a left-parenthesis x plus y right-parenthesis equals italic a x plus italic a y 2nd Row 1st Column Blank 2nd Column normal upper B Baseline 2 colon left-parenthesis a plus b right-parenthesis x equals italic a x plus italic b x 3rd Row 1st Column Blank 2nd Column normal upper B Baseline 3 colon left-parenthesis italic a b right-parenthesis x equals a left-parenthesis italic b x right-parenthesis 4th Row 1st Column Blank 2nd Column normal upper B Baseline 4 colon 1 x equals x left-parenthesis 1 is the unit element right-parenthesis period EndLayout

      The prototypical examples of vector spaces are n-dimensional vectors and rectangular matrices over a field K:

      (4.4)StartLayout 1st Row 1st Column Blank 2nd Column x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis left-parenthesis x Subscript j Baseline element-of upper K comma j equals 1 comma ellipsis comma n right-parenthesis 2nd Row 1st Column Blank 2nd Column y equals left-parenthesis y 1 comma ellipsis comma y Subscript n Baseline right-parenthesis left-parenthesis y Subscript j Baseline element-of upper K comma j equals 1 comma ellipsis comma n right-parenthesis 3rd Row 1st Column Blank 2nd Column x comma y element-of upper K Superscript n Baseline 4th Row 1st Column Blank 2nd Column left-parenthesis x plus y right-parenthesis equals left-parenthesis x 1 plus y 1 comma ellipsis comma x Subscript n Baseline plus y Subscript n Baseline right-parenthesis 5th Row 1st Column Blank 2nd Column normal lamda x equals left-parenthesis normal lamda x 1 comma ellipsis comma normal lamda x Subscript n Baseline right-parenthesis comma normal lamda element-of upper K period EndLayout

      For matrices:

      (4.5)StartLayout 1st Row 1st Column Blank 2nd Column m 1 equals left-parenthesis a Subscript italic i j Baseline right-parenthesis comma m 2 equals left-parenthesis b Subscript italic i j Baseline right-parenthesis comma 1 less-than-or-equal-to i less-than-or-equal-to m comma 1 less-than-or-equal-to j less-than-or-equal-to n 2nd Row 1st Column Blank 2nd Column m 1 plus m 2 equals left-parenthesis a Subscript italic i j Baseline plus b Subscript italic i j Baseline right-parenthesis 3rd Row 1st Column Blank 2nd Column normal lamda m 1 equals left-parenthesis normal lamda a Subscript italic i j Baseline right-parenthesis comma normal lamda element-of upper K period EndLayout

      We now define an important non-negative real-valued function on a vector space V called a norm. It has the following properties:

      (4.6)StartLayout 1st Row 1st Column Blank 2nd Column double-vertical-bar x double-vertical-bar greater-than-or-equal-to 0 semicolon double-vertical-bar x double-vertical-bar equals 0 i f f x equals 0 left-parenthesis if and only if x equals 0 right-parenthesis 2nd Row 1st Column Blank 2nd Column double-vertical-bar normal lamda x double-vertical-bar equals StartAbsoluteValue normal lamda EndAbsoluteValue double-vertical-bar x double-vertical-bar 3rd Row 1st Column Blank 2nd Column double-vertical-bar x 1 plus x 2 double-vertical-bar less-than-or-equal-to double-vertical-bar x 1 double-vertical-bar plus double-vertical-bar x 2 double-vertical-bar left-parenthesis x 1 comma x 2 element-of upper V comma normal lamda element-of upper K right-parenthesis period EndLayout

      Some examples of norms for two-dimensional vectors are:

StartLayout 1st Row 1st Column Blank 2nd Column x equals left-parenthesis x 1 comma x 2 right-parenthesis comma x element-of normal double struck upper R squared 2nd Row 1st Column Blank 2nd Column double-vertical-bar x double-vertical-bar equals StartRoot x 1 squared plus x 2 squared EndRoot 3rd Row 1st Column Blank 2nd Column double-vertical-bar x double-vertical-bar equals max left-parenthesis StartAbsoluteValue x 1 EndAbsoluteValue comma StartAbsoluteValue x 2 EndAbsoluteValue right-parenthesis period EndLayout

      The following norms for vectors and matrices are used in applications:

      (4.7)StartLayout 1st Row 1st Column Blank 2nd Column Euclidean left-parenthesis l 2 right-parenthesis norm double-vertical-bar x double-vertical-bar Subscript 2 Baseline equals left-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts x Subscript j Superscript 2 Baseline right-parenthesis Superscript one half Baseline 2nd Row 1st Column Blank 2nd Column l 1 norm double-vertical-bar x double-vertical-bar Subscript 1 Baseline equals sigma-summation Underscript j equals 1 Overscript n Endscripts StartAbsoluteValue x Subscript j Baseline EndAbsoluteValue 3rd Row 1st Column Blank 2nd Column l Subscript infinity Baseline norm double-vertical-bar x double-vertical-bar Subscript infinity Baseline equals max Underscript 1 less-than-or-equal-to j less-than-or-equal-to n Endscripts StartAbsoluteValue x Subscript j Baseline EndAbsoluteValue period EndLayout

      (4.8)StartLayout 1st Row 1st Column Blank 2nd Column upper L 1 norm colon double-vertical-bar upper A double-vertical-bar Subscript 1 Baseline equals max Underscript 1 less-than-or-equal-to j less-than-or-equal-to n Endscripts left-parenthesis sigma-summation Underscript i equals 1 Overscript n Endscripts StartAbsoluteValue a Subscript italic i j Baseline EndAbsoluteValue right-parenthesis 2nd Row 1st Column Blank 2nd Column upper L Subscript infinity Baseline norm colon double-vertical-bar upper A double-vertical-bar Subscript infinity Baseline equals max Underscript 1 less-than-or-equal-to i less-than-or-equal-to n Endscripts left-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts StartAbsoluteValue a Subscript italic i j Baseline EndAbsoluteValue right-parenthesis period EndLayout

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper D 1 colon d left-parenthesis x comma y right-parenthesis greater-than-or-equal-to 0 semicolon d left-parenthesis x comma y right-parenthesis equals 0 left right double arrow x equals y 2nd Row 1st Column Blank 2nd Column Blank 3rd Column upper D 2 colon d left-parenthesis x comma y right-parenthesis equals d left-parenthesis y comma x right-parenthesis 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper D 3 colon d left-parenthesis x comma y right-parenthesis less-than-or-equal-to d left-parenthesis x comma z right-parenthesis plus d left-parenthesis z comma y right-parenthesis left-parenthesis italic triangle inequality right-parenthesis period EndLayout

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