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## Main Question or Discussion Point

Hi, I was just reading about Orthogonal complements.

I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V.

I then proved that the intersection of W and its orthogonal complement equals 0.

However, I am wondering if the union of W and its orthogonal complement equals V?

Can anyone please answer that, and if so, can you give a proof?

Thanks.

-xfunctionx-

I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V.

I then proved that the intersection of W and its orthogonal complement equals 0.

However, I am wondering if the union of W and its orthogonal complement equals V?

Can anyone please answer that, and if so, can you give a proof?

Thanks.

-xfunctionx-