Functionals->functions of infinite variables?

  • Thread starter Thread starter HomogenousCow
  • Start date Start date
  • Tags Tags
    Infinite Variables
AI Thread Summary
Functionals can be viewed as functions of an infinite number of Taylor coefficients, suggesting they may resemble normal functions that map sets of reals to other sets of reals. However, not all variable functions possess a Taylor expansion, and even if they do, the expansion may only be valid in certain domains. This raises questions about the nature of functionals, especially in spaces like L2, which includes discontinuous functions and those lacking derivatives. The discussion highlights the complexities of defining functionals in relation to traditional functions. Ultimately, the relationship between functionals and functions remains nuanced and context-dependent.
HomogenousCow
Messages
736
Reaction score
213
If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.
 
Mathematics news on Phys.org
HomogenousCow said:
If we think of a functional as a function of the infinite number of taylor coefficients of the variable function, aren't they then just normal functions, a map between a set of reals to another set of reals.

The "variable function" may not have a Taylor expansion, or if it does for part of its domain, the expansion may not hold for the rest. Simple example, L2 includes discontinuous functions, functions without derivatives, etc.
 
Thread 'Video on imaginary numbers and some queries'

Thread 'Video on imaginary numbers and some queries'

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »
Thread 'Unit Circle Double Angle Derivations'

Thread 'Unit Circle Double Angle Derivations'

Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
View full post »

Similar threads

B Are All Imaginable Integer Series Necessarily Infinite?
Replies
17
Views
2K
B Difference between functions and variables
Replies
13
Views
2K
How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?
Replies
23
Views
3K
I Gradient descent algorithm and optimizers
Replies
5
Views
2K
B What is the notation for denoting constants in a function?
Replies
2
Views
1K
I Can you transform a countably infinite set to an uncountable one?
Replies
15
Views
2K
B The terminology "G is only a function of...."
Replies
3
Views
1K
I Alternating Harmonic Numbers are cool, spread the word!
Replies
4
Views
2K
B Understanding exponentiation and logarithms together
Replies
12
Views
2K
I Finding the Largest (or smallest) Value of a Function - given some constant symmetric errors
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top