I am aware that a Frechet space $V$ is a topological vector space whose topology is induced by countably many seminorms $\{ \lVert \cdot \rVert\_k \}$ such that


1. it must be Hausdorff
2. it must be complete


In the [relevant Wikipedia entry](https://en.wikipedia.org/wiki/Fr%C3%A9chet_space)
it is stated that when constructing a Frechet space, one needs to check that for any sequence $\{ x\_n \}$ Cauchy with respect to each seminorm $\lVert \cdot \rVert\_k$, there exists some $x \in V$ such that $x\_n \to x$ w.r.t. each seminorm.


However, I am quite confused here in that, what if one only finds out that there exists $x^k \in V$ for "each" $\lVert \cdot \rVert\_k$ such that $\lVert x\_n-x^k \rVert\_k \to 0$ as $n \to \infty$? That is, there is no guarantee that $x^k = x^{k'}$ for $k \neq k'$.


I tried to resolve this issue by the Hausdroff condition, but cannot find a way out. Could anyone please help me?



Confusion with the definition of a Frechet space regarding completeness and uniqueness of a limit
Consider the space $X$ of sequences $x=(x\_k)\_k \in \mathbf{C}^{\mathbf N}$ with finite support, and let $\|x\|\_k = |x\_k|$. $X$ is clearly not a Fréchet space (one not so elementary reason is that a Fréchet space cannot have countably infinite dimension by Baire's theorem). However, it is easy to see that your condition holds, simply because $\mathbf{C}$ is complete.


