- #1

SithsNGiggles

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## Homework Statement

Let [itex]A[/itex] and [itex]B[/itex] be two orthogonal subspaces of an inner product space [itex]V[/itex]. Prove that [itex]A\cap B= \{ 0\}[/itex].

## Homework Equations

## The Attempt at a Solution

I broke down my proof into two cases:

Let [itex]a\in A, b\in B[/itex].

Case 1: Suppose [itex]a=b[/itex]. Then [itex]\left\langle a,b \right\rangle = \left\langle a,a \right\rangle = 0[/itex], which implies [itex]a=b=0[/itex]. Thus [itex]0 \in A\cap B[/itex].

Case 2: Suppose [itex]a \not= b[/itex]. Then [itex]b \not\in A \wedge a \not\in B[/itex], so [itex]a,b \not\in A \cap B[/itex]. This implies [itex]A \cap B = \emptyset[/itex].

Therefore [itex]A\cap B = \{ 0\}[/itex].My main question is if my second case works. It took me quite some time to convince myself that it was, but now I'm doubting myself again. Thanks