Topologies on spaces of linear maps

Topologies on spaces of linear maps tell us which ways we can think of linear maps and how they are related. A topology is like a set of guidelines or rules that tell us if two linear maps are 'similar'. For example, if they have the same shape, size, or 'direction' then they might be said to be similar or related to each other. So the topology on a space of linear maps tells us how two linear maps might be related, and which ones we should consider to be similar to each other.