Proving the Continuity and Derivative of Distributions: A Scientific Approach

[squenshl]

Homework Statement

How do I prove that if \phi₁, \phi₂ \in D (D is the space of test functions), then \phi₁ + \lambda\phi₂ \in D, (\lambda \in R) also if f is continuous show that the derivative of the distribution defined by f(x)H(x) is f'(x)H(x) + f(0)\delta(x).

Homework Equations

The Attempt at a Solution

For the first proof is it just the same as proving something is a subspace.

 

[arkajad]

First: state exactly the definition of the space of test functions.
Second: state exactly the definition of the derivative of the distribution.

Use both definitions. But they must be written precisely.

 

[squenshl]

D is the space of test functions which consists of all possible smooth functions on R with contact support, so since \phi is smooth with contact support, then the addition of \phi₁ & \phi₂ must be as well (\lambda is just a real number).

 

[squenshl]

If t is a distribution, then the derivative is defined as:
for every \phi \in D, < t' , \phi > = < t , -\phi' > = -< t , \phi' >.

 

[arkajad]

Then the trick goes like this:<\phi,(fH)'>=-<\phi',fH>=-<f\phi',H>=
-<(f\phi)'-f'\phi,H>=<f\phi,H'>+<\phi,f'H>=
<\phi,f'H>+<f\phi,\delta>=...

 

[squenshl]

It's just using the product rule & the fact that H' = \phi(0), thanks heaps.