I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please provide any examples?
Some thoughts:
A complete space is where all Cauchy sequences converge.
A normed vector space is a vector space (say, over $\mathbb{R})$ on some norm $N$ (which is a function that maps $N\to\mathbb{R}$), where the norm obeys the triangle inequality, the norm of a vector is non-negative, and if you have a scalar being multiplied by a vector, you can factor the scalar out, but it'll have absolute value braces.
I'm not really sure what is needed in order to have an unnormed vector space (perhaps the vector space necessarily needs to be infinite dimensional?). Perhaps something really weird like the zero space?
Thanks for any insight.
Best Answer
While your question could have multiple answers, perhaps the closest to what you are looking for is the notion of a non-metrizable vector space.
In the general setting of topological vector spaces, we consider (as one might guess from the name) vector spaces endowed with a topology so that we can discuss ideas like the continuity of linear operators. Normed vector spaces are examples of topological vector spaces where the topology is induced by a given norm.
A non-metrizable vector space is a topological vector space whose topology does not arise from any metric. These are rather common in functional analysis. For example, if $X$ is a Banach space, then the weak-* topology on $X^*$ is never metrizable unless $X$ is finite-dimensional. Another family of examples are locally convex spaces, a natural generalization of Banach spaces, which are not metrizable unless their topology is generated by a countable collection of seminorms that separate points.