I was inspired by this flowchart of mathematical sets and wanted to try and visualize it, since I internalize math best in that way. This is what I've come up with so far:

Is there anything that I'm missing, or that is incorrectly marked? For example, where exactly should I insert a box for Fréchet Spaces? And, is it safe to say that Normed Vector Spaces are a *proper* subset of the intersection between Locally Convex Spaces and Metric Spaces (or is it the entire intersection?)

**Edit:**
Thank you, everyone, for your input. Obviously no single diagram is going to encapsulate the entirety of functional analysis, geometry, and topology (not to mention the myriad of algebraic structures I've ignored, as some of you have pointed out.) As someone who does a lot of analysis, I would often find myself going back to Wikipedia or my textbooks to re-read the definitions of the various spaces and sets I am working with. I just wanted something that could help me keep a lot of these ideas straight in my head; and was pretty and useful to glance at. I think I've settled on my final version (for now.) In summary, here is a quick bullet list of the labeled components of the diagram:

**Topological Spaces**: sets with a notion of what is "open" and "closed".**Vector Spaces**: sets with operations of "addition" and "(scalar) multiplication".**Topological Vector Spaces**: "addition" and "multiplication" are*continuous*in the topology.**Metric Spaces**: sets that come with a way to measure the "distance" between two points, called a*metric*; the topology is generated by this metric.**Locally Convex Spaces**: sets where the topology is generated by translations of "balls" (*balanced*,*absorbent*,*convex*sets); do not necessarily have a notion of "distance".**Normed Vector Spaces**: sets where the topology is generated by a norm, which in some sense is the measure of a vector's "length". A norm can always generate a metric (measure the "length" of the difference of two vectors), and every normed space is also locally convex.**Fréchet Spaces**: a set where the topology is generated by a translation-invariant metric; this metric*doesn't*necessarily have to come from a norm. All Fréchet spaces are*complete*metric spaces (meaning that if elements of a sequence get arbitrarily "close", then the sequence must converge to an element already in the space.)**Banach Spaces**: a set that is a complete metric space, where the metric is defined in terms of a norm.**Inner Product Spaces**: sets with a way to measure "angles" between vectors, called an*inner product*. An inner product can always generate a norm, but the space may or may not be complete with respect to this norm.**Hilbert Spaces**: an inner product space that is*complete*with respect to this induced norm. Any inner product space that is incomplete (called a "pre-Hilbert Space") can be completed to a Hilbert space.**Manifold**: a set with a topology that locally "looks like" Euclidean space. Any manifold can be turned into a metric space.