Let V be a vector space (not necessarily finite dimensional) over a field $\Bbb F$. The dual space of V is the $\Bbb F$-vector space $V^* = L(V, \Bbb F)$.
Prove that as $\Bbb R$-vectors spaces, $(\Bbb R^{ < \infty })^* \cong \Bbb R^{\infty}$ where * denote the dual space.
We have that $\beta = (v_1,v_2,\cdots )$ where $v_1= (1,0,\cdots) $ and each $v_i=(0,\cdots, i, 0, \cdots ) \forall i \in \Bbb N$ i belive this is a basis for $\Bbb R^{ < \infty } $ Which is in this definition at least usually means an infinite vectorspace with finite support like the polynomials of any degree each has a finite number but there are infinite of them.
We have from the definition that $(\Bbb R^{ < \infty })^*= \mathcal L (\Bbb R^{ < \infty },\Bbb R)$
We have that $L:\Bbb R^{<\infty} \to \Bbb R$
I believe that we want $\phi: (\Bbb R^{ < \infty })^* \to \Bbb R^{\infty}$ to be defined in the following way.
$\phi(L)=(L(v_1), L(v_2), \cdots )= (\Bbb R, \Bbb R, \Bbb R, \cdots )=(a_1, a_2, \cdots )$ s.t each $a_i \in \Bbb R $
Now i want to show i have a vector space homomorphism.
We have that $\alpha \in \Bbb R $ and $\alpha (\phi(L))=\alpha (L(v_1), L(v_2), \cdots )= \alpha (a_1,a_2,\cdots) = (\alpha (a_1), \alpha (a_2),\cdots)= \phi(\alpha \space L)$ so scalar multiplication.
We have that if $(a_1,a_2,\cdots)=(b_1,b_2,\cdots )$ implies that $(L(v_1), L(v_2), \cdots ) = (G(v_1), G(v_2), \cdots ) $ this implies that $L(v_1)=G(v_1)$, $L(v_2)=G(v_2), \cdots $, $L(v_i)=G(v_i) \space \space \forall i \in \Bbb N$ this implies that $\phi(L)=\phi (G) $ this is kind of odd and im not sure i showed the right thing but this might be one to one.
I am not really sure what two addition looks like its on two linear functionals right like showing that $\phi ( L+G) = \phi (L) + \phi (G) $ assuming that is the right thing to show i think the following is correct.
Consider $\phi ( L+G)= ( L+G)(v_1), ( L+G)(v_2), \cdots ) $ but L and G are linear functionals! so we have that $( L+G)(v_1), ( L+G)(v_2), \cdots )= (L(v_1)+G(v_1), L(v_2) +G(v_2) ,\cdots )= \phi (L) + \phi (G) $
My problem is onto is a ? and something feels not quite right about one to one...
Edit to comment:
A perfect example of such a thing that is $\Bbb R^{ < \infty }$ is $\mathcal P(\Bbb F) $
a function $ p: \Bbb F \to \Bbb F$ is called a polynomial with coefficients in $\Bbb F $ if there exists $a_0, \cdots , a_m \in \Bbb F $ s.t $p(z)=a_0+a_1 z + \cdots + a_m z^m$ for all $z\in \Bbb F $
Edit2 regarding possible duplicate: Yes that is the same problem as i am asking but the answer posted there is significantly less than what i have posted as my question and in no way imaginable answers my question.