Mathematicians: What are the important definitions in your field?
For example, in topology the notions of compactness and of Hausdorff space are crucial (among many others).
Or in complexity theory the notion of NP-completeness.
And so on.
Can you list the ones in your mathematical field?
Non-practicing-mathematicians: this time just listen what the practicing mathematicians say.
This is an exercise, for me, in learning which concepts working mathematicians of fields different from mine consider to be important for their own field.
PS. I will consider Theoretical Computer Scientists and Theoretical Physicists for this question. And I am also willing to be more general, without any specific example in mind.
@MartinEscardo@mathstodon.xyz - in real analysis, the definition of 'Lebesgue integration' is incredibly important. This relies on the definition of Lebesgue integrable function f:βββ, and thence on the definition of measurable function, and thence on the definition of Lebesgue measurable set, and thence on the definition of Borel set. But for most analysts, integration is the main point of all those more fundamental definitions.
In functional analysis and quantum physics, I'd say the definition of 'Hilbert space' is incredibly important!
In algebraic geometry, I'd say the definitions of 'ample line bundle' and 'coherent sheaf' are incredibly important.
In homotopy theory, the definition of 'simplicial object' is incredibly important.
I'm picking definitions where it's not obvious right away why those definitions will turn out to be so central.
@vnikolov@ieji.de @MartinEscardo@mathstodon.xyz - I think the idea of 'bounded operator' is kind of obvious compared to the definition of 'Banach space', and in quantum mechanics the deep and important definition is that of 'self-adjoint operator', not necessarily bounded.
In short, I don't think 'bounded operator' is a concept with magical power, while 'Hilbert space' and 'self-adjoint operator' do have magical power.
Martin Escardo
