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Let $V = \bigoplus_{n\ge 0} V^n$ be a graded vector space over some field $k$. This grading gives it a filtration $F^pV := \bigoplus_{n\ge p}V^n$, using which we may form the associated graded vector space $\text{gr}(V) := \bigoplus_{p\ge 0} F^pV/F^{p+1}V$.

There's an obvious map $V\rightarrow\text{gr}(V)$ induced by the isomorphisms $\alpha_n : V^n\cong F^nV/F^{n+1}V$, and this seems to be an isomorphism to me.

On the other hand, in the first half of page 4 of this book the author seems to imply that this is only an isomorphism when each $V^n$ is finite dimensional.

I don't see why this is condition is necessary.

Can someone either explain what I'm missing, or explain how I might be misreading the text?

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