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There is a nice visual representation of mathematical spaces from this post:

enter image description here

I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but according to wikipedia an inner product is one:

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

How would one draw a vector space in this image?

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Any normed space is also a vector space. However, you do not need to define a norm on a vector space, meaning that all vector spaces are a union of the "normed space" rectangle and another rectangle that is outside the big "topological space" rectangle.

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