Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subspaces which are disjoint.

I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction to $K_0$ is $0$, and whose restriction to $K_1$ is $1$.

The construction of a continuous function with the above properties is an instance of the usual Urysohn's lemma (and can be written down explicitely in terms of the metric on $V$). But I don't know how to mollify continuous functions on Fréchet spaces….

Note: The same question where $V$ is a Banach space also interests me, and would be sufficient for my purposes.

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint.

I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction to $K_0$ is $0$, and whose restriction to $K_1$ is $1$.

The construction of a continuous function with the above properties is an instance of the usual Urysohn's lemma (and can be written down explicitely in terms of the metric on $V$). But I don't know how to mollify continuous functions on Fréchet spaces….

Note: The same question where $V$ is a Banach space also interests me, and would be sufficient for my purposes.

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subspaces which are disjoint.

I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction to $K_0$ is $0$, and whose restriction to $K_1$ is $1$.

The construction of a continuous function with the above properties is an instance of the usual Urysohn's lemma (and can be written down explicitely in terms of the metric on $V$). But I don't know how to mollify continuous functions on Fréchet spaces...

Note: The same question where $V$ is a Banach space also interests me, and would be sufficiencient for my purposes.

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subspaces which are disjoint.

I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction to $K_0$ is $0$, and whose restriction to $K_1$ is $1$.

The construction of a continuous function with the above properties is an instance of the usual Urysohn's lemma (and can be written down explicitely in terms of the metric on $V$). But I don't know how to mollify continuous functions on Fréchet spaces….

Note: The same question where $V$ is a Banach space also interests me, and would be sufficient for my purposes.