- #1

JD_PM

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- TL;DR Summary
- I want to understand what I am missing in the generalization I made for the direct sum when ##3## subspaces are involved.

Given two subspaces ##U_1, U_2##, I understand the concept of direct sum

$$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$

Where ##W## is a subspace of ##V##.

I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following.

$$ W= U_1 \oplus U_2 \oplus U_3 \iff U_1 \cap U_2 = \{ 0 \}, U_1 \cap U_3 = \{ 0 \}, U_2 \cap U_3 = \{ 0 \}, U_1 + U_2 + U_3 = W $$

It does not seem to have the same structure that for the statement with ##k## subspaces

\begin{align*}

W= U_1 \oplus U_2 \oplus ... \oplus U_k \iff& U_i \cap \left(U_1 + ... + U_{i-1} + U_{i+1} + ... + U_k\right) = \{ 0 \} \\

&U_1 + U_2 + ... + U_k = W

\end{align*}

In particular, the issue lies on the intersection statement. Might you explain why my thought is faulty? I should be able to find a counterexample once I see it :)

Thanks!

$$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$

Where ##W## is a subspace of ##V##.

I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following.

$$ W= U_1 \oplus U_2 \oplus U_3 \iff U_1 \cap U_2 = \{ 0 \}, U_1 \cap U_3 = \{ 0 \}, U_2 \cap U_3 = \{ 0 \}, U_1 + U_2 + U_3 = W $$

It does not seem to have the same structure that for the statement with ##k## subspaces

\begin{align*}

W= U_1 \oplus U_2 \oplus ... \oplus U_k \iff& U_i \cap \left(U_1 + ... + U_{i-1} + U_{i+1} + ... + U_k\right) = \{ 0 \} \\

&U_1 + U_2 + ... + U_k = W

\end{align*}

In particular, the issue lies on the intersection statement. Might you explain why my thought is faulty? I should be able to find a counterexample once I see it :)

Thanks!