If any linear subspace is closed then is it in general true that it is invariant under the differentiation operator? Could any one help me to prove it formally?
For example in the vector space $C[0,1]$, set of all polynomials $\mathbb{P}_n(\mathbb{R})$ say of deg $n$ forms a subspace and it is closed subspace since it is a finite dimensional. Now under differentiation operator $D:C[0,1]\to C[0,1]$ clearly $D(\mathbb{P}_n(\mathbb{R}))\subsetneq \mathbb{P}_n(\mathbb{R})$
The subspace spanned by the one element $\sin(x)$ is closed, but it is not invariant under differentiation.