Prove dual space has the direct sum decomposition

[jpcjr]

I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.

Any direction would be greatly appreciated!

Mike

Homework Statement

Let V be a vector space,

Let W₁, ..., W_k be subspaces of V, and,

Let V_j = W₁ + ... + W_j-1 + W_j+1 + ... + W_k.

Suppose that V = W₁ $\oplus$ ... $\oplus$ W_k.

Prove that the dual space V* has the direct-sum decomposition V* = V^o₁ $\oplus$ ... $\oplus$ V^o_k.

Homework Equations

See above.

The Attempt at a Solution

Again, I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.

 

[micromass]

What is the ${}^\circ$ in $V_j^\circ$?

 

[jpcjr]

What is the " ^o " in V^o_j?

Definition. If V is a vector space over the field F and S is a subset of V, the annihilator of S is the set S^o of linear functionals f on V such that f(α) = 0 for every α in S.
.
.
.

 

[micromass]

OK. You must prove

$$V^*=V_1^\circ \oplus ... \oplus V_n^\circ$$

What does that mean?? What is the definition of a direct sum? What is it you need to check?

 

[jpcjr]

Thank you!

By the skin of my teeth, some help from you, and the grace of God, I received the best grade I could have expected in Linear Algebra.

Thanks, again!

Joe