Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $T:E\to X$, such that $T(E)=U$.

Even though operator ranges are clearly not necessarily closed, they have many special properties, e.g. those highligted in Reference request: Baire's theorem for operator ranges.

In fact I was led to this post because I am after an answer to the following:

Question. Is every operator range a Baire space in the relative topology?

The title of the post mentioned above is tantalizingly close but unfortunately it doesn't offer any answer to my question, neither I was able to find it anywhere else. Any ideas?

In case this question turns out to be senstitive to the kind of Banach spaces considered, I am really interested in Hilbert-operator-ranges, that is, subspaces of Hilbert spaces which coincide with the range of a bounded linear map defined on a Hilbert space.

Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $T:E\to X$, such that $T(E)=U$.

Even though operator ranges are clearly not necessarily closed, they have many special properties, e.g. those highligted in Reference request: Baire's theorem for operator ranges.

In fact I was led to this post because I am after an answer to the following:

Question. Is every operator range a Baire space (in the sense that the intersection of countably many open dense sets is dense) in the relative topology?

The title of the post mentioned above is tantalizingly close but unfortunately it doesn't offer any answer to my question, neither was I able to find it anywhere else. Any ideas?

In case this question turns out to be senstitive to the kind of Banach spaces considered, I am really interested in separable-Hilbert-operator-ranges, that is, subspaces of separable Hilbert spaces which coincide with the range of a bounded linear map defined on a separable Hilbert space.

Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $T:E\to X$, such that $T(E)=U$.

Even though operator ranges are clearly not necessarily closed, they have many special properties, e.g. those highligted in Reference request: Baire's theorem for operator ranges.

In fact I was led to this post because I am after an answer to the following:

Question. Is every operator range a Baire space in the relative topology?

The title of the post mentioned above is tantalizingly close but unfortunately it doesn't offer any answer to my question, neither I was able to find it anywhere else. Any ideas?

Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $T:E\to X$, such that $T(E)=U$.

Even though operator ranges are clearly not necessarily closed, they have many special properties, e.g. those highligted in Reference request: Baire's theorem for operator ranges.

In fact I was led to this post because I am after an answer to the following:

Question. Is every operator range a Baire space in the relative topology?

The title of the post mentioned above is tantalizingly close but unfortunately it doesn't offer any answer to my question, neither I was able to find it anywhere else. Any ideas?

In case this question turns out to be senstitive to the kind of Banach spaces considered, I am really interested in Hilbert-operator-ranges, that is, subspaces of Hilbert spaces which coincide with the range of a bounded linear map defined on a Hilbert space.