Definition: Let $F$ be a field. Let $V$ and $V^\backsim$ be linear algebra over $F$. The algebras $V$ and $V^\backsim$ are isomorphic if $\exists \backsim :V\to V^\backsim$ defined by $\backsim (\alpha)=\alpha^\backsim$ such that $\backsim$ is bijective, $(c\cdot \alpha+\beta)^\backsim$ $=c\cdot \alpha^\backsim +\beta^\backsim$ and $(\alpha \beta)^\backsim =\alpha^\backsim \beta^\backsim$, $\forall \alpha ,\beta \in V$, $\forall c\in F$.
If $F$ is a field containing an infinite number of distinct elements, the mapping $f\to f^\backsim$ is an isomorphism of the algebra of polynomials over $F$ onto the algebra of polynomial function over $F$.
My attempt: Let $F[x]=\mathrm{span}(\{x^n|n\geq 0\})$ and $V=\{p:F\to F|\text{ }p(t)=\sum_{i=0}^nc_i \cdot t^i,\text{ }\forall t\in F$, for some $c_i\in F\}$. If $f$ $=(f_0,f_1,..,f_n,0,0,…)$ $=\sum_{i=0}^n f_i\cdot x^i$, then $f^\backsim:F\to F$ such that $f^\backsim (t)$ $=\sum_{i=1}^n f_i\cdot t^i$, $\forall t\in F$. So $f(t)$ $= \sum_{i=1}^n f_i\cdot t^i$ $=f^\backsim (t)$. Define $\backsim :F[x]\to V$ such that $\backsim (f)=f^\backsim$. It’s easy to check $(c\cdot f+g)^\backsim =c\cdot f^\backsim +g^\backsim$, proof is similar to part (i) of theorem 2 section 4.2. Let $f,g\in F[x]$. Then $(f\cdot g)^\backsim (t)$ $=(f\cdot g)(t)$ $=f(t)\cdot g(t)$ $=f^\backsim (t)\cdot g^\backsim (t)$ $=(f^\backsim \cdot g^\backsim) (t)$, $\forall t\in F$. Thus $(f\cdot g)^\backsim =f^\backsim \cdot g^\backsim$. Let $p\in V$. Then $\exists c_0,c_1,…,c_n\in F$ such that $p(t)=\sum_{i=0}^nc_i \cdot t^i$, $\forall t\in F$. Define $L:V\to F[x]$ such that $L(p)=(c_0,c_1,..,c_n,0,0,…)$, $\forall p\in V$. We need to show $\backsim \circ$ $L=\text{id}_{V}$ and $L$ $\circ \backsim$ $=\text{id}_{F[x]}$. Let $p$ be a polynomial function with $c_0,c_1,…,c_n$ coefficient. Then $L(p)=(c_0,c_1,..,c_n,0,0,…)$. So $\backsim \circ$ $L(p)$ $=\backsim (L(p))$ $=\backsim((c_0,c_1,..,c_n,0,0,…))$ $=p$. Thus $\backsim \circ$ $L=\text{id}_{V}$. Let $f=(f_0,f_1,..,f_n,0,0,…)\in F[x]$. Then $L$ $\circ \backsim (f)$ $=L(\backsim (f))$ $=L(f^\backsim)$ $=f$. Thus $L$ $\circ \backsim$ $=\text{id}_{F[x]}$. Hence $\backsim$ is bijective map. So $F[x]\cong V$. Is my proof correct?
Hoffman’s proof explicitly showed injectivity. Another way to show $\backsim (f)=\backsim (g)$$\implies$$f=g$ is given in Axler book. Que: Where did we use $|F|=\infty$ fact?