Intro
I am currently learning category theory and the first example for a universal property from that script is the one of the direct sum in the category of vector spaces. My question is actually less concerned with the universal property itself but the definition that the author is using:
Direct sum
General definiton:
Let $\ \{ V_j \}_{j ∈ J} \ $ be a family of vector spaces over $ \mathbb F $.
$ \bigoplus_{j \in J} V_{j} = \{ (v_j)_{j ∈ J} \ | \ v_j = 0 \quad \text{for all but finitely many} \} \qquad (A * ) $
Problem
Our prof defined the sum as follows:
$ \sum_{j \in J} V_{j} = \{ \sum_{j \in J} u_j \in V \ | \ \forall \ j \in J : u_j \in V_{j} \} $
And we get the direct sum iff:
$ V_{j} \bigcap \sum_{i=1 \\ i ≠ j} V_{i} = \{0_V\} $.
where $ V_j $ are subspaces of $ V $ and $ J $ is finite. I am aware that the definition in the finite case must be different bc in $(A*)$ there are infinite spaces. But i would assume that there is some comprehensible extension from the finite case to the infinite.
I'd expect that if I take def. $ (A*) $ with finite $ J $ there should be an isomorphism between that def. and the one from our prof.
What I read into def $(A*)$ is that this is a set of all sequences of vectors in $V$, e. g. of the form : $((v_1^{1}, v_1^{2})^{\top}, (v_2^{1}, v_2^{2})^{\top}) $ where $ v_1 = (v_1^{1}, v_1^{2})^{\top} $ , $ v_2 = (v_2^{1}, v_2^{2})^{\top} ∈ \mathbb F^2 $.
How could someone say that this is the same? The shape of vectors is already totally different and are in $ \mathbb F^{2 \times 2} $ whereas based on the def from our prof these vectors would be in $ \mathbb F^2 $ with a form $ (v_1^{1} + v_2^{1}, v_1^{2} + v_2^{2})^{\top} $.
But $ \mathbb F^{2 \times 2} $ and $ \mathbb F^2 $ are not isomorphic. The best I can do is to sum up each row so we get an element in $ \mathbb F^{2 \times 1} $ and from here I can get an isomorphism to $ \mathbb F^2 $.
But then the way back is impossible.
I must misunderstand def. $ (A*) $. Please help!
It seems to me that you are mixing up the "internal" definition of direct sum and the "external" one:
$(A*)$ is stated in terms of an abstract family of vector spaces $\{V_j\mid j\in J\}$, or rather in terms of a functor $J \to \sf Vect$ where $\sf Vect$ is the category of vector spaces. The external direct sum $\bigoplus V_j$ is their coproduct.
An essential feature of the direct sum as an operation on vector spaces is that, if $J$ is finite, the same object $\bigoplus_j V_j$ has the universal property of a product as well.
the "internal" direct sum of vector subspaces of a given vector space $V$ is what your professor defined: the subspace generated by the set theoretic union of all the subspaces $V_j$ (although there is the subtlety that the sum $\sum_j v_j$ doesn't really have a meaning if $J$ is infinite... but I think you got the definition right, and just worded it imprecisely)
The external definition is what's usually used in category theory, and the subspace $V_j\le V$ is usually thought as a vector space of independent existence from that of $V$.