Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$?

It is obviously satisfied if $W$ is finite-dimensional. We think that we can prove it for the case when $\phi$ is onto, $V$ is a Fréchet space and $W$ is a countable product of finite dimensional spaces.

Do you know any other generality that ensures it?

Thanks a lot

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$?

It is obviously satisfied if $W$ is finite-dimensional.

Do you know any reasonable generality that ensures it?

Thanks a lot

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$.

It is obviously satisfied if $W$ is finite-dimensional. We think that we can prove it for the case when $\phi$ is onto, $V$ is a Frechet space and $W$ is a countable product of finite dimensional spaces.

Do you know any other generality that insure it?

Thanks a lot

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$?

It is obviously satisfied if $W$ is finite-dimensional. We think that we can prove it for the case when $\phi$ is onto, $V$ is a Fréchet space and $W$ is a countable product of finite dimensional spaces.

Do you know any other generality that ensures it?

Thanks a lot