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# areEqual(DualSpace,DualSpace) -- approximate equality of dual spaces

- Function: areEqual
- Usage:
    `b = areEqual(A,B)`
- Inputs:
  - A, an instance of the type DualSpace,
  - B, an instance of the type DualSpace,
- Optional inputs:
  - Projective => ..., default value false, determine if solutions are equal
  - Tolerance => ..., default value .000001, the tolerance of a numerical computation
- Outputs:
  - b, a Boolean value,

## Description

Two dual spaces are approximately equal if the have (approximately) the same base point(DualSpace) and the linear spaces spanned by the differential operators are equal approximately.

    i1 : R = CC[x,y];

    i2 : A = dualSpace(matrix{{y^2,x^2+x*y}},point{{1,1}})

    o2 = | y2 x2+xy |

    o2 : DualSpace

    i3 : B = dualSpace(matrix{{x^2+x*y+y^2,y^2+0.00000001}},point{{1,1+0.00000001}})

    o3 = | x2+xy+y2 y2+1e-8 |

    o3 : DualSpace

    i4 : b = areEqual(A,B)

    o4 = true

## Ways to use this method:

- areEqual(DualSpace,DualSpace) -- approximate equality of dual spaces

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NAGtypes/doc-NAGtypes.m2:984:0.