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nice point. One way of looking at it, I think, is that you have just two Brownian motions, so in a sense your space is just 2 dimensional. Thus, as long as $V$ and $V_1$ are not "linearly dependent", you're spanning the space, and you're done, and it doesn't really matter what $V_1$ you're choosing. Now, this is of course a very hand-waving argument, in ...

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