Is the Dirac Delta a function or a generalized distribution?

It's a function from the vector space of smooth functions to the corresponding fields

>>9125024
Distributions are functions. I don't understand why people get triggered when you call the [math]\delta[/math] a function. Have they ever opened a functional analysis book?

>>9125067
It's not a function.
It's not even necessary.

Check out the Riemann–Stieltjes integral.

https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral

Discontinuities in the integrator perform discrete samplings of the integrand with a weight equal to the size of the "jump".

A unit step function for the integrator does the same thing as the delta.

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>>9125220
Functional Analysis (Walter Rudin).
Now crop me an alternative definition of distributions.

>>9125024
Lil peep

Leavvveeeee me alone

>>9125024
define function. and im not being pedantic, it realy does depend. the dirac delta is a functional, which is a function from a vector space to a its underlying field. even if you use the underlying set using the forgetful functor U its still a function from one set to another. if you define function as it is defined in calculus, as a subset of the cartesian product of the doman and codomain with some additional constraints, then its not a function though.

>>9125934
>[..] then its not a function though.
You honestly don't know what you are talking about.
But then you're like freshman so whatever.

>>9125938
its not though, to be a function as usually defined in calc the codomain must be in R, infinity isnt in R.

>>9125934
I just realized I made an error, its NOT a function on the vector space itself, but its still is a function on the underlying set of the vector space. It is also a function on the vector space of function over the base set. I accidentally combined those 2 things into one on my post

>>9125892
https://en.wikipedia.org/wiki/Spede_Pasanen

>>9125943
Codomain must be in R only for real valued function. For general function it's codomain can be an arbitrary set

>>9125934
A function f:A->B is a relation from the set A to the set B, such that for all a in A there exists one and only one b in B such that afb (a is related to b by f)

>>9125024
distribution

it is undefined at 0

those are the same thing OP

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>>9125024
Neither

>>9127322
>>9126450
yes, in set theory in general its also a function, but not in calc/analyses where the domain and codomain must be R^n

>>9128379
Unironically kill yourself.

>>9127404
>it is undefined at 0
[math]\langle \delta,\varphi\rangle=\varphi (0)[/math] you useless bag of meat.
[math]0[/math] is the zero vector hence
[math]\langle \delta,0\rangle=0[/math].
Do not post here if you haven't ever opened a Functional Analysis textbook you negroid.

>>9128492
fine, K^n where K can be R or C if you want to be anal about it.

it's a functional whatever that means I never call it a function