Gelfand triple
Okay kiddo, let me try to explain gelfand triple to you. Have you ever gone to a park and seen a seesaw? A seesaw has two sides, one higher than the other, and a pivot in the middle. Imagine that we have three different playgrounds, each with a seesaw that looks a bit different.
In the first playground, the seesaw only has one side, that's the higher side. In the second playground, the seesaw has both sides, but no pivot. And in the third playground, the seesaw has both sides and also a pivot.
Now imagine that we can assign certain things to each playground. To the first playground, we assign something called the test functions. These are special types of functions that are very smooth and can be used to measure other functions. To the second playground, we assign the space of all functions, which can be good, bad, or somewhere in between. And finally, to the third playground, we assign something called the tempered distributions, which are sort of like a more powerful version of the test functions.
These three things together form what is called a Gelfand triple, named after a mathematician named Israel Gelfand. The test functions and the tempered distributions help us to understand and measure the functions in the space of all functions. It's like the seesaw - without the pivot, the two sides don't interact with each other. But in the Gelfand triple, the test functions and tempered distributions help to balance and measure the functions in the space.
So that's a Gelfand triple for you, kiddo! It might seem a bit complicated, but it's really just like playground seesaws.
In the first playground, the seesaw only has one side, that's the higher side. In the second playground, the seesaw has both sides, but no pivot. And in the third playground, the seesaw has both sides and also a pivot.
Now imagine that we can assign certain things to each playground. To the first playground, we assign something called the test functions. These are special types of functions that are very smooth and can be used to measure other functions. To the second playground, we assign the space of all functions, which can be good, bad, or somewhere in between. And finally, to the third playground, we assign something called the tempered distributions, which are sort of like a more powerful version of the test functions.
These three things together form what is called a Gelfand triple, named after a mathematician named Israel Gelfand. The test functions and the tempered distributions help us to understand and measure the functions in the space of all functions. It's like the seesaw - without the pivot, the two sides don't interact with each other. But in the Gelfand triple, the test functions and tempered distributions help to balance and measure the functions in the space.
So that's a Gelfand triple for you, kiddo! It might seem a bit complicated, but it's really just like playground seesaws.