When do curves exist in infinite-dimensional submanifolds?

Question

Let me explain the motivation for my question by talking about the finite-dimensional situation. Let's say we have a $d$-dimensional $C^\infty$ manifold $M$ embedded smoothly in $\mathbb R^n$. We fix some subspace $H\subset \mathbb R^n$ that intersects $M$ transversally, so that $H\cap M$ is a smooth submanifold of $M$. Let $x\in H\cap M$ and identify the tangent space $T_xM$ with a subspace of $\mathbb R^n$, so that $T_x(H\cap M)=H\cap T_xM$. In particular, there exist curves passing through $x$ in the direction of all the vectors on $H\cap T_xM$. The existence of these curves is what I need, but in the infinite dimensional setting.

My setting is as follows: we have a separable topological vector space $V$ and a dense subspace $W$. The kinematic tangent space to $W$ is all of $V$; I am able to find one explicit example of a curve $c_{p,v}$ completely contained in $W$ going in the direction of each vector $v$ in $V$. Now, take a subspace $X\subset V$ and consider $X\cap W$. I would then expect the kinematic tangent space to $X\cap W$ to coincide with $X$. Unfortunately, most of the time my curve $c_{p,v}$ in $W$ passing by a point of $p\in X\cap W$ in the direction of a vector $v\in X$ is not contained in $X\cap W$ and instead intersects this set only at $p$.

Question. Is it obvious that there is still a curve going through $p$ in the direction $v$, or do I have to prove it?

Thanks a lot in advance!

Answer 1

Take $V = L^2(M)$ with its usual Hilbert space topology and $W = C^\infty(M)$. Take $X = \mathrm{span}\{f\}$ for some non-smooth function. Then $X \cap W = \{0\}$, hence the kinematic tangent space of $X \cap W$ is zero as well, and not equal to $X$.

Your situation is quite strange and the example in the beginning of your post is not really an analog, as $W$ is not closed, and hence not a submanifold, as it is not complemented.