Let $X$ and $Y$ be two topological spaces and $f:X\to Y$ a function. It is well-known that continuity of $f$ implies sequentially continuity, while the reverse is in general only true if $X$ is first countable.
Now, if $U\subset\mathbb{R}^{d}$ is some open set, then one defines the space of test functions $\mathcal{D}(U)$ to be the set $C^{\infty}_{c}(U)$ equipped with the canonical LF-space topology. As a topological space, $\mathcal{D}(U)$ is not metrizable and also not first countable. The space of distributions is then usually defines to be the topological dual space$$\mathcal{D}^{\prime}(U):=(\mathcal{D}(U))^{\prime}$$By definition, the topological dual space is the set of linear continuous functionals.
Question: In virtually all textbooks I have seen, a distribution is defined to be a linear function $u:\mathcal{D}(U)\to\mathbb{C}$ which is sequentially continuous, i.e. $\varphi_{n}\to 0$ implies $u(\varphi_{n})\to 0$. However, then the space of distributions is not the topological dual space $(\mathcal{D}(U))^{\prime}$, but something weaker right? Since for the space $\mathcal{D}(U)$, sequentially continuity does not imply continuity. Am I missing something?