In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that
$$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) \to (X,\mathscr{T})$$
$$\iota_+: (x,y) \mapsto x+y$$
and $$\iota_\cdot: (\mathbb{K},\mathscr{U}) \times (X, \mathscr{T}) \to (X,\mathscr{T})$$
$$\iota_\cdot: (\alpha,x) \mapsto \alpha x$$
are continuous.
Now, my question may seem rather dim, but I have not seen an explicit reason for Why we require these mappings to be continuous. What does this ensure, and why is it of importance. Naturally this is true in Banach and Hilbert spaces, since a map between norm spaces is equivalent to it being bounded by some $C$ so we may just take $C= \alpha$ and $C= n\max\{x_i\}_{i=1}^{n}$.