If $V$ is a vector space and $P_1, \dots, P_k\in\mathcal{L}(V)$ are projections satisfying

1. $P_iP_j=0$ if $i\neq j$
2. P_1+\dots+ P_k=I_V

then $V=V_1\oplus\dots\oplus V_k$ for $V_i=\text{Ran}P_i$.
Hence, direct sum decompositions correspond to families of projections satisfying the two properties.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

Related

If $V$ is a vector space and $P_1, \dots, P_k\in\mathcal{L}(V)$ are projections satisfying

1. $P_iP_j=0$ if $i\neq j$
2. P_1+\dots+ P_k=I_V

then $V=V_1\oplus\dots\oplus V_k$ for $V_i=\text{Ran}P_i$.
Hence, direct sum decompositions correspond to families of projections satisfying the two properties.

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

Related