Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the usual topology of of uniform convergence on compact subsets of all partial derivatives.

It tuns out that sometimes $f^*$ can be extended to larger spaces of generalized functions. More precisely, fix a closed $\mathbb{R}_{>0}$-invariant subset $\Lambda\subset T^*Y\backslash 0$ of the cotangent bundle with removed zero section. Let $C^{-\infty}_\Lambda(Y)$ denote the space of generalized functions with the wave front set contained in $\Lambda$. This space $C^{-\infty}_\Lambda(Y)$ is equipped with some standard locally convex linear topology (see e.g. Ch. 6 in the book "Geometric Asymptotics" by Guillemin and Sternberg).

Let us assume that the map $f$ is transversal to $\Lambda$ in the sense that for any $x\in X$ if $(f(x),\eta)\in \Lambda$ then $(df_x^*)(\eta)\ne 0$. Then one defines a ("natural") linear map

$$f^*\colon C^{-\infty}_\Lambda(Y)\to C^{-\infty}(X)\,\,\,\,\,\,\,\,\,\, (1)$$

extending the usual pull-back on smooth functions (see the above mentioned book. Another very good reference is Hormander's ”The analysis of linear partial differential operators, I”; see especially Theorem 8.2.4.)

The point is that in the above literature **the map (1) is proven to be sequentially continuous**, namely it maps convergent sequences to convergent ones.

**QUESTION. Is the pull-back map (1) topologically continuous?**

The difference between usual topological continuity (e.g. continuity in the usual sense of maps of topological spaces) and sequential continuity seems to me to be quite subtle.

**Edit:** Definition of topology on $C^{-\infty}_\Lambda(X)$.
Covering $X$ by open charts and using
partition of unity we may assume that $X= \mathbb{R}^n$. For any $N\in\mathbb{N},
\phi \in C^\infty_c(\mathbb{R}^n)$, and any closed $\mathbb{R}_{>0}$-invariant
subset $V\subset \mathbb{R}^n$ such that
$$\Lambda\cap (supp(\phi)\times V)=\emptyset$$
let us define the semi-norm on $C^{-\infty}_\Lambda(\mathbb{R}^n)$ by
$$||u||_{\phi,N,V}=sup_{\xi\in V}|\xi|^N|\widehat{\phi u}(\xi)|,$$
where $\hat F$ denotes the Fourier transform of the function $F$.
Then one equips $C^{-\infty}_\Lambda(X)$ with the weakest locally
convex topology which is stronger than the weak topology on
$C^{-\infty}(X)$ and such that all semi-norms
$\{||\cdot||_{\phi,N,V}\}$ are continuous.

Topological Vector Spaces and Distributions. $\endgroup$ – TaQ Oct 2 '13 at 20:11