Defining isomorphism between dual spaces

Question

I'm having troubles trying to define an isomorphism between (V $\bigoplus$ W)* and V* $\bigoplus$ W*; where V and W are vector spaces over a field K; V*, W* are respectively the dual spaces; and $\bigoplus$ is the direct sum.
I tried to define this function $T$: (V $\bigoplus$ W)* $\rightarrow$ V* $\bigoplus$ W* such that: $T$($\alpha$+$\beta$)=$\alpha$+$\beta$ .
But I don't know if this is an isomorphism.

Answer 1

It's not clear what $T(\alpha + \beta) = \alpha + \beta$ means. In particular, if $\alpha: V \to \Bbb K$ and $\beta : W \to \Bbb K$, then how exactly is $(\alpha + \beta)(u)$ defined for an arbitrary $u \in V \oplus W$?

One isomorphism which does work is to take $$ T[\alpha \oplus \beta](v + w) = \alpha(v) + \beta(w) $$ where the above holds for arbitrary $v \in V$ and $w \in W$.

First, note that the definition of $\oplus$ means that the map $T[\alpha \oplus \beta]$ is well defined. That is: if $v + w = v' + w'$ for $v,v' \in V$ and $w,w' \in W$, then we necessarily have $T[\alpha \oplus \beta](v + w) = T[\alpha \oplus \beta](v' + w')$, which accounts for the potential ambiguity of this definition. Next, note that the definition of $\oplus$ also implies that the above determines $T[\alpha \oplus \beta]$ on all of $V \oplus W$.

I claim that it is pretty clear that $T$ is linear, but you should convince yourself that this is true. From there, it suffices to show that $T$ is both injective (one-to-one) and surjective (onto).

Answer 2

I can only think of one thing to do.

Coproduct comes equipped with canonical injections $i_j: U_i \hookrightarrow \bigoplus_{i \in I} U_i$.

In this case, try the maps induced by $f \mapsto f \circ i_j$.

In particular, given a map $[f:U_1 \oplus U_2 \to k] \in (U_1\oplus U_2)^*$, send it to $(f \circ i_1,f \circ i_2) \in U_1^* \oplus U_2^*$