I Definition of Limit for vector fields

Kashmir
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Apostol defines limit for vector fields as
> ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)##
means that :
##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0##

Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##
 
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Scalars? vectors ?

##\ ##
 
Kashmir said:
Apostol defines limit for vector fields as
> ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)##
means that :
##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0##

Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##
You can, after you accept the definition. I think going from f(x)= b to f(x)-b is 0, is obvious, but going from lim(x->a) to lim (||x-a|| ->0) is not.
There are certainly functions where it's important that || f(x) - b || < epsilon for all the x with || x-a || < delta.
 
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Kashmir said:
Apostol defines limit for vector fields as
> ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)##
means that :
##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0##

Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##
Say ##\lim_{x \to a} f(x)=b## means, that for each neighborhood ##V_b## of ##b## there is a neighborhood ##U_a## of ##a## such that ##f(x)\in V_b## whenever ##x\in U_a\,.## If we further assume that we are dealing with topological vector spaces, then this is equivalent to ##\lim_{x-a \to 0}(f(x)-b)=0\,.## It also implies that every neighborhood contains an ##\varepsilon -##ball. Thus we can replace all neighborhoods with ##\varepsilon -##balls, i.e. the norm convergence. And metric spaces are topological vector spaces.
 
BvU said:
Scalars? vectors ?

##\ ##
I couldn't understand. Could you please explain what you mean?
 
willem2 said:
You can, after you accept the definition. I think going from f(x)= b to f(x)-b is 0, is obvious, but going from lim(x->a) to lim (||x-a|| ->0) is not.
There are certainly functions where it's important that || f(x) - b || < epsilon for all the x with || x-a || < delta.
"lim(x->a) to lim (||x-a|| ->0)" because if x->a the magnitude of difference vector ||x-a|| will tend to zero.
Is that ok?
 
fresh_42 said:
Say ##\lim_{x \to a} f(x)=b## means, that for each neighborhood ##V_b## of ##b## there is a neighborhood ##U_a## of ##a## such that ##f(x)\in V_b## whenever ##x\in U_a\,.## If we further assume that we are dealing with topological vector spaces, then this is equivalent to ##\lim_{x-a \to 0}(f(x)-b)=0\,.## It also implies that every neighborhood contains an ##\varepsilon -##ball. Thus we can replace all neighborhoods with ##\varepsilon -##balls, i.e. the norm convergence. And metric spaces are topological vector spaces.
Thank you for your reply. I can't understand it right now. It's beyond my learning. Hopefully I'll come back to it after studying some more. :)
 
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