Return to Question

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder how the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Edit : OK, I need to change the condition $T_n(A) \subset T_{n+1}(A)$ by the one that

"there exists a projection $P_{n,n'} : T_n(V_m) \to T_{n'}(V_m)$ for all $n \geq n' \geq m$ such that $P_{n', k} \circ P_{n,n'}= P_{n,k}$ for $n \geq n' \geq k$ and $P_{n,n}=Id$." Also, I further asssume that the image of each $T_n$ consists of smooth functions as well. In this case, Is the above union a finite-dimensional space consisting of smooth functions, so that I can use "any" $L^p$ norm on it?

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder what the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Edit : OK, I need to change the condition $T_n(A) \subset T_{n+1}(A)$ by the one that

"there exists a projection $P_{n,n'} : T_n(V_m) \to T_{n'}(V_m)$ for all $n \geq n' \geq m$ such that $P_{n', k} \circ P_{n,n'}= P_{n,k}$ for $n \geq n' \geq k$ and $P_{n,n}=Id$." Also, I further asssume that the image of each $T_n$ consists of smooth functions as well. In this case, Is the above union a finite-dimensional space consisting of smooth functions, so that I can use "any" $L^p$ norm on it?

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder how the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Edit : OK, I need to change the condition $T_n(A) \subset T_{n+1}(A)$ by the one that

"there exists a projection $P_{n,n'} : T_n(V_m) \to T_{n'}(V_m)$ for all $n \geq n' \geq m$ such that $P_{n,n'} \circ P_{n', k}= P_{n,k}$ for $n \geq n' \geq k$ and $P_{n,n}=Id$." Also, I further asssume that the image of each $T_n$ consists of smooth functions as well. In this case, Is the above union a finite-dimensional space consisting of smooth functions, so that I can use "any" $L^p$ norm on it?

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder how the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Edit : OK, I need to change the condition $T_n(A) \subset T_{n+1}(A)$ by the one that

"there exists a projection $P_{n,n'} : T_n(V_m) \to T_{n'}(V_m)$ for all $n \geq n' \geq m$ such that $P_{n', k} \circ P_{n,n'}= P_{n,k}$ for $n \geq n' \geq k$ and $P_{n,n}=Id$." Also, I further asssume that the image of each $T_n$ consists of smooth functions as well. In this case, Is the above union a finite-dimensional space consisting of smooth functions, so that I can use "any" $L^p$ norm on it?

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder how the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further that each $V_n$ consists of smooth functions.

Consider a collection of invertible linear mappings $\{T_n\}_{n=1}^\infty$ defined on each $V_n$ such that $T_n(A) \subset T_{n+1}(A)$ for all $A \subset V_n$.

Now, if we fix some $m \in \mathbb{N}$, then the image $T_n(V_m)$ has the same dimension as $V_m$ for all $n \geq m$. Now, I wonder how the vector space \begin{equation} \bigcup_{n=1}^\infty T_n(V_m) \end{equation} looks like. It certainly looks like finite-dimensional, but is it possible to say more about this space? For example, does it still consist of smooth functions?

Also, if $v \in V_m$, in what sense does $T_n(v)$ converge in $\bigcup_{n=1}^\infty T_n(V_m)$ as $n \to \infty$?

I think LF or LB spaces are relevant topics, but cannot find some concrete information to apply to this case. Could anyone please help me?

Edit : OK, I need to change the condition $T_n(A) \subset T_{n+1}(A)$ by the one that

"there exists a projection $P_{n,n'} : T_n(V_m) \to T_{n'}(V_m)$ for all $n \geq n' \geq m$ such that $P_{n,n'} \circ P_{n', k}= P_{n,k}$ for $n \geq n' \geq k$ and $P_{n,n}=Id$." Also, I further asssume that the image of each $T_n$ consists of smooth functions as well. In this case, Is the above union a finite-dimensional space consisting of smooth functions, so that I can use "any" $L^p$ norm on it?