Show that $$M = \{ x = (x_k)_{k \in \mathbb{N}} \in \ell^p \mid x_{2k} = 0 \ \forall k \in \mathbb{N} \}$$ is a closed subspace of $\ell^p$. Further, $\text{codim}(M) = \infty$, $\ell^p \cong M$, and $\ell^p / M \cong \ell^p$.
To be very frank, this quest is also available here (Quotient space of $l^p$ that isometrically isomorphic to $l^p$) , but as a beginner I don’t understand the condition that is provided in $M$, I know about $\ell^p$ spaces and understand the idea of isometrically isomorphism. But can’t understand how to proceed?
“I mainly want the proof of the last part, I.e quotient space is isometrically isomorphic to the main space “
I have search a lot to understand and tried to understand from that answered questions but I can’t go any further. So I ultimately decide to ask the question here again.
Thank you for generous help. Thank you so much.