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.