Isomorphic quotient spaces

Question

I am unable to come up with a quotient space and subspace for the following question.

let $X$ and $Y$ be Banach spaces and let $T: X\to Y$ be an onto bounded linear operator. Show that $Y$ is isomorphic to a quotient space of $X$ and that $Y$ is isomorphic to a subspace of $X$.

Since $Y$ may not be a subspace of $X$, I am having difficulty coming up with the quotient space and subspace.

Your help is very much appreciated. Thank you.

Answer 1

Hint: Show that$$\begin{array}{ccc}X/\ker T&\longrightarrow&Y\\x+\ker T&\mapsto&T(x)\end{array}$$is an isomorphism.