Counting Balls in $L^2_m[0,1]$
$begingroup$
Setup and Thoughts to Date
Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a separating sequence $(f_n in L^2_m[0,1])_{n in mathbb{N}}$ such that (here's I've used Riez to identify $L^2_m[0,1]$ with its topological dual)
$$
d^{star}(x,y)triangleq sum_{n in mathbb{N}}frac{f_n(x-y)}{2^n},
$$
defines a metrizes $B$ under the weak topology on $L^2_m[0,1]$.
The Banach-Alaoglou theorem, implies that $B$ is a compact metric space under this metric. Therefore the open cover
$$
mathcal{U}triangleq left{
overline{Ballleft(x;frac1{2}right)}
right}_{x in B},
$$
admits a finite subcover.
Question
My question is, what is the cardinality of this finite sub-cover?
real-analysis metric-spaces hilbert-spaces weak-lp-spaces
asked Jan 21 at 9:57
N00berN00ber
19511
$endgroup$
add a comment |
$begingroup$
Setup and Thoughts to Date
Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a separating sequence $(f_n in L^2_m[0,1])_{n in mathbb{N}}$ such that (here's I've used Riez to identify $L^2_m[0,1]$ with its topological dual)
$$
d^{star}(x,y)triangleq sum_{n in mathbb{N}}frac{f_n(x-y)}{2^n},
$$
defines a metrizes $B$ under the weak topology on $L^2_m[0,1]$.
The Banach-Alaoglou theorem, implies that $B$ is a compact metric space under this metric. Therefore the open cover
$$
mathcal{U}triangleq left{
overline{Ballleft(x;frac1{2}right)}
right}_{x in B},
$$
admits a finite subcover.
Question
My question is, what is the cardinality of this finite sub-cover?
real-analysis metric-spaces hilbert-spaces weak-lp-spaces
asked Jan 21 at 9:57
N00berN00ber
19511
$endgroup$
1
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02
add a comment |
$begingroup$
Setup and Thoughts to Date
Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a separating sequence $(f_n in L^2_m[0,1])_{n in mathbb{N}}$ such that (here's I've used Riez to identify $L^2_m[0,1]$ with its topological dual)
$$
d^{star}(x,y)triangleq sum_{n in mathbb{N}}frac{f_n(x-y)}{2^n},
$$
defines a metrizes $B$ under the weak topology on $L^2_m[0,1]$.
The Banach-Alaoglou theorem, implies that $B$ is a compact metric space under this metric. Therefore the open cover
$$
mathcal{U}triangleq left{
overline{Ballleft(x;frac1{2}right)}
right}_{x in B},
$$
admits a finite subcover.
Question
My question is, what is the cardinality of this finite sub-cover?
real-analysis metric-spaces hilbert-spaces weak-lp-spaces
asked Jan 21 at 9:57
N00berN00ber
19511
$endgroup$
Setup and Thoughts to Date
Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a separating sequence $(f_n in L^2_m[0,1])_{n in mathbb{N}}$ such that (here's I've used Riez to identify $L^2_m[0,1]$ with its topological dual)
$$
d^{star}(x,y)triangleq sum_{n in mathbb{N}}frac{f_n(x-y)}{2^n},
$$
defines a metrizes $B$ under the weak topology on $L^2_m[0,1]$.
The Banach-Alaoglou theorem, implies that $B$ is a compact metric space under this metric. Therefore the open cover
$$
mathcal{U}triangleq left{
overline{Ballleft(x;frac1{2}right)}
right}_{x in B},
$$
admits a finite subcover.
Question
My question is, what is the cardinality of this finite sub-cover?
real-analysis metric-spaces hilbert-spaces weak-lp-spaces
real-analysis metric-spaces hilbert-spaces weak-lp-spaces
asked Jan 21 at 9:57
N00berN00ber
19511
asked Jan 21 at 9:57
N00berN00ber
19511
asked Jan 21 at 9:57
N00berN00ber
19511
asked Jan 21 at 9:57
N00berN00ber
19511
asked Jan 21 at 9:57
N00berN00ber
19511
19511
1
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02
add a comment |
1
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02
1
1
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02
add a comment |
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1
$begingroup$
Can you use geometric series arguments to compute it?
$endgroup$
– AIM_BLB
Jan 22 at 10:14
$begingroup$
Not sure how that would work? Compute the limiting number?
$endgroup$
– N00ber
Jan 22 at 14:02