I'm familiar with the advice by Geoffrey Hinton, "To deal with a 14-dimensional space, visualize a 3-D space and say ‘fourteen’ to yourself very loudly. Everyone does it." I'm happy with this to visualize high-dimensional spaces and even countably-infinite-dimensional spaces. Vectors are just points in this space.
Are there tricks to visualize uncountably-infinite-dimensional spaces? Since it's uncountable, I don't know how to enumerate axes so that I can just picture a few of them. e.g. how do I think about boundedness, completeness, compactness of subsets of continuous functions (without relying on algebraic manipulations alone)?
An element of an uncountably infinite-dimensional vector space can be visualized by arranging its components with respect to the appropriate basis, numbered by real numbers, along the real line. E.g., a plot of some function defined on the set $R^1$ (say, sine, exponent, etc., and not necessarily continuous) will be such a visualization.