There’s no notion of continuity in linear algebra per se. It’s not part of the definition of a vector space. But a finite dimensional vector space over the reals is isomorphic to a Euclidean space of the same dimension, and so we usually think of such spaces as Euclidean. (We’ll only going to consider real vector spaces in this post.) And there we have a notion of distance, a norm, and hence a topology and a way to say whether a function is continuous.
In finite dimensional Euclidean space, linear functions are continuous. You can put a different norm on a Euclidean space than the one it naturally comes with, but all norms give rise to the same topology and hence the same continuous functions. (This is useful in numerical analysis where you’d like to look at a variety of norms. The norms give different analytical results, but they’re all topologically equivalent.)
In an infinite dimensional normed space, linear functions are not necessarily continuous. If the dimension of a space is only a trillion, all linear functions are continuous, but when you jump from high dimension to infinite dimension, you can have discontinuous linear functions. But if you look at this more carefully, there isn’t a really sudden change.
If a linear function is discontinuous, its finite dimensional approximations are continuous, but the degree of continuity is degrading as dimension increases. For example, suppose a linear function stretches the nth basis vector by a factor of n. The bigger n gets, the more the function stretches in the nth dimension. As long as n is bounded, this is continuous, but in a sense it is less continuous as n increases. The fact that the infinite dimensional version is discontinuous tells you that the finite dimensional versions, while technically continuous, scale poorly with dimension. (See practical continuity for more discussion along these lines.)
A Banach space is a complete normed linear space. Finite dimensional normed spaces are always complete (i.e. every sequence in the space converges to a point in the space) but this might not happen in infinite dimensions.
Duals and double duals
In basic linear algebra, the dual of a vector space V is the space of linear functionals on V, i.e. the set of linear maps from V to the reals. This space is denoted V*. If V has dimension n, V* has dimension n, and all n-dimensional spaces are isomorphic, so the distinction between a space and its dual seems pedantic. But in general a Banach space and its dual are not isomorphic and so its easier to tell them apart.
The second dual of a vector space, V** is the dual of the dual space. In finite dimensional spaces, V** is naturally isomorphic to V. In Banach spaces, V is isomorphic to a subset of V**. And even when V is isomorphic to V**, it might not be naturally isomorphic to V**. (Here “natural” means natural in the category theory sense of natural transformations.)