codimension of complementary subspace

Asked Jun 27 '20 at 06:35

Active Jun 27 '20 at 06:35

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X is a linear space, and Y and N are complementary subspaces of X, i.e. $X = N \oplus Y.$ Show that when N has finite codimension, $dim Y = codim N$.

This is an exercise from Lax's functional analysis. My question is that why it supposes N has finite codimension.

asked Jun 27 '20 at 06:35

St. Peter

1

If you define the codimension as the of the quotient $X/N$ then $Y$ is isomorphic to this quotient and hence has the same Dimension. – Jochen Jun 27 '20 at 12:51
Thanks for your reply, but I knew that before. I don't know why N must have finite codimension.@Jochen – St. Peter Jun 27 '20 at 13:50
This also holds for infinite codimension. – Jochen Jun 27 '20 at 13:51
@Jochen You enlightening comment has helped me solve my question. Thank you so much! – Analyst May 20 '23 at 19:52

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