a notation for convergeence.
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Suppose ${f_n}$ is a sequence of complex functions and $|f_n(x)-f(x)|to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|to 0$, does it show that the convergence is uniformly convergence?
calculus sequences-and-series uniform-convergence sequence-of-function
edited Nov 19 at 18:41
Niklas
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asked Nov 19 at 18:07
mathrookie
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Suppose ${f_n}$ is a sequence of complex functions and $|f_n(x)-f(x)|to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|to 0$, does it show that the convergence is uniformly convergence?
calculus sequences-and-series uniform-convergence sequence-of-function
edited Nov 19 at 18:41
Niklas
2,178720
asked Nov 19 at 18:07
mathrookie
724512
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose ${f_n}$ is a sequence of complex functions and $|f_n(x)-f(x)|to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|to 0$, does it show that the convergence is uniformly convergence?
calculus sequences-and-series uniform-convergence sequence-of-function
edited Nov 19 at 18:41
Niklas
2,178720
asked Nov 19 at 18:07
mathrookie
724512
Suppose ${f_n}$ is a sequence of complex functions and $|f_n(x)-f(x)|to 0$ for all $x$. If we put "for all $x$" behind the $|f_n(x)-f(x)|to 0$, does it show that the convergence is uniformly convergence?
calculus sequences-and-series uniform-convergence sequence-of-function
calculus sequences-and-series uniform-convergence sequence-of-function
edited Nov 19 at 18:41
Niklas
2,178720
asked Nov 19 at 18:07
mathrookie
724512
edited Nov 19 at 18:41
Niklas
2,178720
asked Nov 19 at 18:07
mathrookie
724512
edited Nov 19 at 18:41
Niklas
2,178720
edited Nov 19 at 18:41
Niklas
2,178720
edited Nov 19 at 18:41
Niklas
2,178720
2,178720
asked Nov 19 at 18:07
mathrookie
724512
asked Nov 19 at 18:07
mathrookie
724512
asked Nov 19 at 18:07
mathrookie
724512
724512
add a comment |
add a comment |
1 Answer
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1
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No. The sentences
$|f_n(x)-f(x)|to0$ for all $x$
- for all $x$, $|f_n(x)-f(x)|to0$
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $epsilon$-$delta$ definition of the convergence, namely you replace $forall epsilon forall x exists delta cdots$ with $forall epsilon exists delta forall x cdots$.
answered Nov 19 at 18:17
Federico
2,649510
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
No. The sentences
$|f_n(x)-f(x)|to0$ for all $x$
- for all $x$, $|f_n(x)-f(x)|to0$
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $epsilon$-$delta$ definition of the convergence, namely you replace $forall epsilon forall x exists delta cdots$ with $forall epsilon exists delta forall x cdots$.
answered Nov 19 at 18:17
Federico
2,649510
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
add a comment |
up vote
1
down vote
accepted
No. The sentences
$|f_n(x)-f(x)|to0$ for all $x$
- for all $x$, $|f_n(x)-f(x)|to0$
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $epsilon$-$delta$ definition of the convergence, namely you replace $forall epsilon forall x exists delta cdots$ with $forall epsilon exists delta forall x cdots$.
answered Nov 19 at 18:17
Federico
2,649510
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
No. The sentences
$|f_n(x)-f(x)|to0$ for all $x$
- for all $x$, $|f_n(x)-f(x)|to0$
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $epsilon$-$delta$ definition of the convergence, namely you replace $forall epsilon forall x exists delta cdots$ with $forall epsilon exists delta forall x cdots$.
answered Nov 19 at 18:17
Federico
2,649510
No. The sentences
$|f_n(x)-f(x)|to0$ for all $x$
- for all $x$, $|f_n(x)-f(x)|to0$
express the same fact. It's just a matter of rewriting the sentence in an equivalent form, which is a feature of English and many other languages.
You pass from pointwise to uniform convergence when you interchange the quantifiers in the $epsilon$-$delta$ definition of the convergence, namely you replace $forall epsilon forall x exists delta cdots$ with $forall epsilon exists delta forall x cdots$.
answered Nov 19 at 18:17
Federico
2,649510
answered Nov 19 at 18:17
Federico
2,649510
answered Nov 19 at 18:17
Federico
2,649510
answered Nov 19 at 18:17
Federico
2,649510
2,649510
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
add a comment |
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
If there is statement:there exists a sequence $f_n$ on $X$ such that $|f_n(ab)-f_n(a)f_n(b)| to 0$ for all $a,b$ in $X$,is the convergence here pointwise or uniformly?
– mathrookie
Nov 20 at 19:50
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
Written like this, the convergence is universally intended to be pointwise. I just means that once you fix values of $a$ and $b$, the sequence you wrote converges to $0$.
– Federico
Nov 21 at 15:35
add a comment |
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