Suppose that we have a finite-dimensional real vector space $V$ and its dual space of linear functionals $V^*$.
Now, suppose that $S$ and $T$ are subspaces of $V$. For now we will require $S$ and $T$ to have trivial intersection, but later we will look at general subspaces.
As these are also vector spaces, we can look at the dual spaces $S^*$ and $T^*$. As $S$ and $T$ are subspaces of $V$, $S^*$ and $T^*$ can be naturally viewed as quotient spaces of $V^*$. In particular, we have that $S^* = V^*/\text{ann}(S)$, where $\text{ann}(S)$ is the annihilator of $S$. This equality is in the sense that for any $f$ in $V^*$ and any $s$ in $S$, we have that $\langle f, s\rangle = \langle f+g, s \rangle$, so that we can identify linear functionals on $S$ with entire cosets of $V$.
Now, the terminology question:
- The space $S \oplus T$ is the "direct sum" of $S$ and $T$.
- The dual space of $(S \oplus T)^*$ is the "______________" of $S^*$ and $T^*$??
What is the __________? Is there a name for this "co-direct sum?"
If $S$ and $T$ have trivial intersection, this is simply a different quotient space whose kernel is the intersection of the kernels of $S^*$ and $T^*$ in $V^*$.
If they don't have trivial intersection then this is very interesting. For instance, suppose we have that $S = T = V$. Then we have that $S\oplus T = V\oplus V$, and in this situation we get that the "co-direct sum" is also equal to the direct sum $S^* \oplus T^* = V^* \oplus V$.
What is the name for this?