Mixing
Does $L_1$ convergence of continuous functions imply pointwise convergence?
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Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?
I'm pretty sure the answer is no, but I'm not aware of a counterexample.
functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
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add a comment |
$begingroup$
Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?
I'm pretty sure the answer is no, but I'm not aware of a counterexample.
functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
$endgroup$
add a comment |
$begingroup$
Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?
I'm pretty sure the answer is no, but I'm not aware of a counterexample.
functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
$endgroup$
Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $xin[0,1]$?
I'm pretty sure the answer is no, but I'm not aware of a counterexample.
functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence
functional-analysis convergence examples-counterexamples lp-spaces pointwise-convergence
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
asked Jan 24 at 2:52
Keshav SrinivasanKeshav Srinivasan
2,33421445
2,33421445
add a comment |
add a comment |
2 Answers
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Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
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The "dancing indicators" counterexample:
$$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$
Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
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1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
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– eepperly16
Jan 24 at 4:28
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
$endgroup$
add a comment |
$begingroup$
Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
$endgroup$
add a comment |
$begingroup$
Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
$endgroup$
Consider $f_n(x) = (-1)^n x^n.$ Then $f_nin C[0,1],$ and $f_n to 0$ in $L^1,$ but $f_n(1)$ does not converge.
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
answered Jan 24 at 4:23
zhw.zhw.
74.2k43175
74.2k43175
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add a comment |
$begingroup$
The "dancing indicators" counterexample:
$$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$
Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
$endgroup$
1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
add a comment |
$begingroup$
The "dancing indicators" counterexample:
$$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$
Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
$endgroup$
1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
add a comment |
$begingroup$
The "dancing indicators" counterexample:
$$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$
Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
$endgroup$
The "dancing indicators" counterexample:
$$chi_{[0, 1/2]}, chi_{[1/2, 1]}, chi_{[0, 1/3]}, chi_{[1/3, 2/3]}, chi_{[2/3, 1]}, chi_{[0, 1/4]}, chi_{[1/4, 2/4]}, ldots$$
Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x in [0,1]$, rather than one that fails to converge pointwise at a single $x$.
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
edited Jan 24 at 7:24
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
answered Jan 24 at 3:05
angryavianangryavian
42.1k23381
42.1k23381
1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
add a comment |
1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
1
1
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
$begingroup$
well, those are not continuous
$endgroup$
– zhw.
Jan 24 at 4:20
1
1
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
$begingroup$
If you modify the indicators to be "trapezoidal", having a linear ramp to the peak value, this example fails to converge pointwise at any point in the interval
$endgroup$
– eepperly16
Jan 24 at 4:28
add a comment |
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