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Michael_1812
Michael_1812
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I thought that this would be a simpe question, and placed it here it at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that a natural map $$ \bigoplus \mathbb{V}_i\longrightarrow\mathbb{V} $$ exists and is a homeomorphism?

Here $\bigoplus \mathbb{V}_i$ is a set of all families $\left(\mathbb{v}_i\right)_{i\in\cal I}$ with $v_i\in {\mathbb V}_i$ and only finitely many non-zero $v_i$. $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

I thought that this would be a simpe question, and placed here it at Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that a natural map $$ \bigoplus \mathbb{V}_i\longrightarrow\mathbb{V} $$ exists and is a homeomorphism?

Here $\bigoplus \mathbb{V}_i$ is a set of all families $\left(\mathbb{v}_i\right)_{i\in\cal I}$ with $v_i\in {\mathbb V}_i$ and only finitely many non-zero $v_i$. $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that a natural map $$ \bigoplus \mathbb{V}_i\longrightarrow\mathbb{V} $$ exists and is a homeomorphism?

Here $\bigoplus \mathbb{V}_i$ is a set of all families $\left(\mathbb{v}_i\right)_{i\in\cal I}$ with $v_i\in {\mathbb V}_i$ and only finitely many non-zero $v_i$. $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

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Michael_1812
Michael_1812
  • 603
  • 1
  • 5
  • 13

I thought that this would be a simpe question, and placed here it at Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that thea natural map $$ \{{\mathbb{V}}_i\}_{i\in{\cal I}}\longrightarrow\mathbb{V} $$$$ \bigoplus \mathbb{V}_i\longrightarrow\mathbb{V} $$ exists and is a homeomorphism?

Here $\bigoplus \mathbb{V}_i$ is a set of all families $\left(\mathbb{v}_i\right)_{i\in\cal I}$ with $v_i\in {\mathbb V}_i$ and only finitely many non-zero $v_i$. $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

I thought that this would be a simpe question, and placed here it at Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that the map $$ \{{\mathbb{V}}_i\}_{i\in{\cal I}}\longrightarrow\mathbb{V} $$ exists and is a homeomorphism? $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

I thought that this would be a simpe question, and placed here it at Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that a natural map $$ \bigoplus \mathbb{V}_i\longrightarrow\mathbb{V} $$ exists and is a homeomorphism?

Here $\bigoplus \mathbb{V}_i$ is a set of all families $\left(\mathbb{v}_i\right)_{i\in\cal I}$ with $v_i\in {\mathbb V}_i$ and only finitely many non-zero $v_i$. $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!

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Michael_1812
Michael_1812
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Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces

I thought that this would be a simpe question, and placed here it at Mathematics Stackexchange. Now have to elevate it to Mathoverflow.

LANGUAGE

TVS = topological vector space. Any subspace of a TVS is a TVS in the induced-topology sense.

DEFINITION

For TVS spaces ${\mathbb{V}}_1\subset\mathbb V$, a TVS subspace ${\mathbb{V}}_2\subset\mathbb V$ is a topological complement of ${\mathbb{V}}_1$ in $\mathbb V$, if $\mathbb V$ is their direct sum both algebraically and topologically. This implies that $$ {\mathbb{V}}_1\oplus {\mathbb{V}}_2\,=\,{\mathbb{V}}~\,, $$ and the following addition map is a homeomorphism: $$ {\mathbb{V}}_1\times{\mathbb{V}}_2\,\longrightarrow\,{\mathbb{V}}~~,\qquad \left(\, v\in{{\mathbb V}}_1\,,~\,v^{\,\prime}\in{{\mathbb V}}_2\,\right)\,\longmapsto~ (v+v^{\,\prime})\in\mathbb V~~.\\ $$

QUESTION A

For a COUNTABLY infinite splitting $$ {\mathbb V}=\bigoplus_{i\in\cal I}{\mathbb V}_i $$ to be not only algebraic but also topological, would it be sufficient to impose the condition that the map $$ \{{\mathbb{V}}_i\}_{i\in{\cal I}}\longrightarrow\mathbb{V} $$ exists and is a homeomorphism? $~~~~~\\\\$

QUESTION B

Would this work also for an uncountable sum (direct integral) of subspaces?

Stated alternatively, can we always be sure that there always exists a necessary measure on the set of these subspaces?

If there is no general answer to question B, can this question be answered for Hilbert spaces?$~~~~~\\\\$

KIND REQUEST

This question is of interest mainly to physicists. Could you please make your answer sufficiently detailed and, if possible, understandable to a layman? Thank you!