nLab algebraic limit field

Contents

Contents

Idea

A field with a notion of a limit of a function that satisfy the algebraic limit theorems.

Definition

Let FF be a Heyting field and a Hausdorff function limit space, where x 1x^{-1} is another notation for the reciprocal function 1x\frac{1}{x}. FF is a algebraic limit field if the algebraic limit theorems are satisfied, i.e. if the limit preserves the field operations:

  • for all elements cSc \in S,

    lim xc0(x)=0\lim_{x \to c} 0(x) = 0
  • for all elements cSc \in S and functions f:SCf:S \to C and g:SCg:S \to C such that

    lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c
    lim xcf(x)+g(x)=lim xcf(x)+lim xcg(x)\lim_{x \to c} f(x) + g(x) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)
  • for all elements cSc \in S and functions f:SCf:S \to C such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c
    lim xcf(x)=lim xcf(x)\lim_{x \to c} -f(x) = -\lim_{x \to c} f(x)
  • for all elements cSc \in S and functions f:SCf:S \to C and g:SCg:S \to C such that

    lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c
    lim xcf(x)g(x)=lim xcf(x)lim xcg(x)\lim_{x \to c} f(x) - g(x) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)
  • for all elements cSc \in S, integers aa \in \mathbb{Z}, and functions f:SCf:S \to C, such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c
    lim xcaf(x)=alim xcf(x)\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)
  • for all elements cSc \in S and aSa \in S, and functions f:SCf:S \to C, such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c
    lim xcaf(x)=alim xcf(x)\lim_{x \to c} a f(x) = a \lim_{x \to c} f(x)
  • for all elements cSc \in S,

    lim xc1(x)=1\lim_{x \to c} 1(x) = 1
  • for all elements cSc \in S and functions f:SCf:S \to C and g:SCg:S \to C such that

    lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c
    lim xcf(x)g(x)=lim xcf(x)lim xcg(x)\lim_{x \to c} f(x) \cdot g(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)
  • for all elements cSc \in S, natural numbers nn \in \mathbb{N}, and functions f:SCf:S \to C, such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c
    lim xcf(x) n=(lim xcf(x)) n\lim_{x \to c} {f(x)}^n = {\left(\lim_{x \to c} f(x)\right)}^n
  • for all elements cSc \in S, and functions f:SCf:S \to C, such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c

    if

    lim xcf(x)#0\lim_{x \to c} f(x) \# 0

    then

    lim xcf(x) 1=(lim xcf(x)) 1\lim_{x \to c} {f(x)}^{-1} = {\left(\lim_{x \to c} f(x)\right)}^{-1}
  • for all elements cSc \in S, and functions f:SCf:S \to C, such that

    lim xcf(x)=c\lim_{x \to c} f(x) = c

    if

    lim xcf(x)#0\lim_{x \to c} f(x) \# 0

    then

    lim xcf(x)f(x) 1=1\lim_{x \to c} f(x) \cdot {f(x)}^{-1} = 1

See also

Last revised on May 4, 2022 at 19:41:09. See the history of this page for a list of all contributions to it.