Proving the Continuity and Derivative of Distributions: A Scientific Approach

[squenshl]

Homework Statement

How do I prove that if $$\phi$$₁, $$\phi$$₂ $$\in$$ D (D is the space of test functions), then $$\phi$$₁ + $$\lambda$$$$\phi$$₂ $$\in$$ 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 $$\phi$$₁ & $$\phi$$₂ 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.