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Let $A\subset \ell^2$ consists of all $x\in \ell^2$ with $|x|_2=1$ which does not belonge to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non empty with a Baire category argument.

I am interested in (some what) distribution and configuration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. Note that density of a subset $A$ at a point $p$ of a metric space with measure $\mu$ is the limit $\frac{\mu(A\cap B_\epsilon)}{\mu (B_{\epsilon})}$ as$\epsilon$ goes to 0.($B_\epsilon$ is the ball around $p$).

But I realized that the measure theory on infinite Banach space is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non-empty with a Baire category argument.

I am interested in (somewhat) distribution and configuration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. Note that density of a subset $A$ at a point $p$ of a metric space with measure $\mu$ is the limit $\frac{\mu(A\cap B_\epsilon)}{\mu (B_{\epsilon})}$ as  $\epsilon$ goes to 0.  ($B_\epsilon$ is the ball around $p$).

But I realized that the measure theory on infinite Banach spaces is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space

Let $A\subset \ell^2$ consists of all $x\in \ell^2$ with $|x|_2=1$ which does not belonge to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non empty with a Baire category argument.

I am interested in (some what) distribution and configuration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. But I realized the measure theory on infinite Banach space is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space

Let $A\subset \ell^2$ consists of all $x\in \ell^2$ with $|x|_2=1$ which does not belonge to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non empty with a Baire category argument.

I am interested in (some what) distribution and configuration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. Note that density of a subset $A$ at a point $p$ of a metric space with measure $\mu$ is the limit $\frac{\mu(A\cap B_\epsilon)}{\mu (B_{\epsilon})}$ as$\epsilon$ goes to 0.($B_\epsilon$ is the ball around $p$).

But I realized that the measure theory on infinite Banach space is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space

Let $A\subset \ell^2$ consists of all $x\in \ell^2$ with $|x|_2=1$ which does not belonge to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non empty with a Baire category argument.

I am interested in (some what) distribution and confuguration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. But I realized the measure theory on infinite Banach space is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space

Let $A\subset \ell^2$ consists of all $x\in \ell^2$ with $|x|_2=1$ which does not belonge to any $\ell^p$ for all $0<p<2$.

Note that $A$ is non empty with a Baire category argument.

I am interested in (some what) distribution and configuration of $A$ and its complement in the unit sphere $S$ of $\ell^2$.

Note that $S\setminus A$ is path connected.

Is $S\setminus A$ a contractible space?

What can be said about connected components of $A$? In particular can one find at least one non constant curve?

Note: I was thinking to a possible density of $A$ in the sphere, or to a possible measure of $A$ as a subset of sphere. But I realized the measure theory on infinite Banach space is a technical matter:

What is known about the Gaussian measure of the unit ball in a Hilbert Space?

What is the 'right' definition of zero measure subsets of Banach spaces?

Invariant probability on a unit ball of a Banach space

https://math.stackexchange.com/questions/75932/measure-on-hilbert-space