Return to Question

A colleague asked me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"

This norm is used in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observed that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space which is isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.

A colleague asked me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"

This interesting norm is introduced in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observed that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space which is isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.

A colleague asked me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"

This norm is used in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observed that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.

A colleague asked me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"

This norm is used in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observed that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space which is isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.

A colleague asks me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$.

This norm is used in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observe that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.

A colleague asked me the following question:

"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"

This norm is used in Continuous mixed p-norm adaptive algorithm for system identification, Hadi Zayyani 2014, IEEE Signal Processing letters, doi:10.1109/LSP.2014.2325495

I observed that this is not a complete norm since it is dominated by $\ell^1$ norm but $\ell^1$ norm is not dominated by this norm. So I am curious about the structure of the completion of this norm.

What is a standard Banach space isomorphic to the completion of this norm on $\ell^1$? Is it a reflexive Banach space?

Note: Since 2 month ago I try to find a possible feature of "continuous field of Banach spaces" but I do not know how to apply this theory to this particular norm. Any suggestion or help is very appreciated.