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Question involving annihilators and solution spaces

britatuni

New member
Aug 9, 2013
2
I was hoping to get help on a question that has been bugging me, I goes like this:

V is a vector space with a dual space V* and U is a subspace of V and W a subspace of V*

The question ask to show that:
'the solution space of W intersected with U' is a subspace of 'the solution space of (W + the annihilator of U)'.


Now, looking at the left hand side I see that an element, 'x', within U must be satisfy f(x)=0 for all functions, 'f', within W.
I realise that the above is barely a start on the question at all. But after looking at eh definitions I just don't see where I am expected to go next.

Please Help!
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
Hello britatuni! I'm not familiar with your definitions of solution space and annihilator. Can you write them for me? English is not my native language and I think I know what is being asked, but I cannot be sure until the terms are defined. :D

Cheers!
 

britatuni

New member
Aug 9, 2013
2
Hi, sorry for the late response since I was on holiday without internet.

I have the solution space of W defined as the set of elements (v) in V that satisfy the condition that f(v) = 0 for every function f in the dual space of W.
And the annihilator is defined as the set of functions (g) in V*such that g(u) = 0 for every element u in the subspace U.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
I was hoping to get help on a question that has been bugging me, I goes like this:

V is a vector space with a dual space V* and U is a subspace of V and W a subspace of V*

The question ask to show that:
'the solution space of W intersected with U' is a subspace of 'the solution space of (W + the annihilator of U)'.


Now, looking at the left hand side I see that an element, 'x', within U must be satisfy f(x)=0 for all functions, 'f', within W.
I realise that the above is barely a start on the question at all. But after looking at eh definitions I just don't see where I am expected to go next.
So it appears that "the solution space of $W\,$" is what I would call the pre-annihilator of $W$, namely the space $W_{\perp} \stackrel {\text{def}}{=} \{x\in X : f(x) = 0\ \forall f\in W\}$. You are asked to show that $W_{\perp} \cap U \subseteq \bigl(W + U^{\perp}\bigr)_{\perp}$ (where $U^{\perp}\stackrel {\text{def}}{=} \{f\in W : f(u)=0\ \forall u\in U\}$ is the annihilator of $U$).

Let $x\in W_{\perp} \cap U$. You correctly say that this implies $f(x) = 0$ for all $f$ in $W$. You are asked to show that $f(x) = 0$ for all $f$ in $W + U^{\perp}$. An element of $W + U^{\perp}$ is by definition the sum of an element in $W$ and an element in $U^{\perp}$. Show that both those elements of $V^*$ must vanish at $x$, and you have completed the proof.

[I think that this problem belongs to linear algebra rather than analysis, so I have transferred it to the Linear and Abstract Algebra section.]
 
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