Question involving annihilators and solution spaces

[britatuni]

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 realize 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]

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]

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]

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 realize 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.]

 

[blue_raver22]

I understand your frustration with this question. It involves some complex concepts and may seem daunting at first. However, with a clear understanding of the definitions and properties involved, we can break it down and solve it step by step.

First, let's define some terms. V is a vector space, which means it is a set of objects that can be added and multiplied by scalars (numbers) to form new objects. V* is the dual space of V, which consists of all linear functions that map elements of V to real numbers. U is a subspace of V, which means it is a subset of V that also satisfies the properties of a vector space. W is a subspace of V*, which means it is a subset of V* that satisfies the properties of a vector space.

Now, the question is asking us to show that the solution space of W intersected with U (denoted as W ∩ U) is a subspace of the solution space of (W + the annihilator of U) (denoted as W + Ann(U)). In order to prove this, we need to understand what these solution spaces represent.

The solution space of W is the set of all vectors in V that satisfy all the linear functions in W. Similarly, the solution space of U is the set of all vectors in V that satisfy all the linear functions in U. The annihilator of U is the set of all linear functions in V* that map all vectors in U to 0. Therefore, the solution space of (W + Ann(U)) is the set of all vectors in V that satisfy all the linear functions in W and all the linear functions in Ann(U).

Now, let's focus on the left-hand side of the equation, W ∩ U. This represents the set of all vectors in V that satisfy all the linear functions in W and all the linear functions in U. Since U is a subspace of V, any vector that satisfies all the linear functions in U must also satisfy all the linear functions in W. Therefore, W ∩ U is a subset of the solution space of W.

Moving on to the right-hand side, W + Ann(U), this represents the set of all vectors in V that satisfy all the linear functions in W and all the linear functions in Ann(U). Since Ann(U) consists of all linear functions that map vectors in U to 0, any vector that satisfies all the linear functions in Ann