Is the Dirac Delta a function or a generalized distribution?
Is the Dirac Delta a function or a generalized distribution?
Angel Baker
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Jeremiah Long
It's a function from the vector space of smooth functions to the corresponding fields
Jaxson Morgan
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?
Cooper James
It's not a function.
It's not even necessary.
Check out the Riemann–Stieltjes integral.
en.wikipedia.org
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.
Hudson Morgan
Functional Analysis (Walter Rudin).
Now crop me an alternative definition of distributions.
Robert Rogers
Lil peep
Leavvveeeee me alone
Ian Cruz
...
Dominic Long
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.
Luke Foster
>[..] then its not a function though.
You honestly don't know what you are talking about.
But then you're like freshman so whatever.
Eli Sullivan
its not though, to be a function as usually defined in calc the codomain must be in R, infinity isnt in R.
Evan Allen
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
Gavin Foster
Jayden Watson
Codomain must be in R only for real valued function. For general function it's codomain can be an arbitrary set
Thomas Nelson
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)
Connor Taylor
distribution
it is undefined at 0
Owen Wright
those are the same thing OP
Benjamin Taylor
Neither
Carter Clark
yes, in set theory in general its also a function, but not in calc/analyses where the domain and codomain must be R^n
John Roberts
Unironically kill yourself.
Evan Jackson
>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.
Sebastian Gomez
fine, K^n where K can be R or C if you want to be anal about it.
Isaac Gutierrez
it's a functional whatever that means I never call it a function