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Let X be a vector space and Y a normed subspace with norm $||_Y$. Let $A : X \rightarrow Y$ be a linear mapping.

(a) Prove that with the prescription $||x|| = ||Ax||_Y$ for every $x \in X$, a norm is defined on the space $X$ if and only if $A$ is an injective mapping.

(b) Suppose $A$ is a bijective linear mapping and $Y$ is a Banach space. Prove that in this case, the space $X$ with the norm defined in part (a) becomes a Banach space as well.

I was able to prove (a), but not (b). However, my attempt:

(b) Now, let's assume $A$ is a bijective linear mapping and $Y$ is a Banach space. To show that $X$ with the norm defined in part (a) becomes a Banach space, we need to show that every Cauchy sequence in $X$ converges to a limit in $X$.

Let $(x_n)$ be a Cauchy sequence in $X$. Since $X$ is a normed space, a Cauchy sequence is also bounded. Since $A$ is bijective, its inverse $A^{-1}$ exists.

Define $y_n = Ax_n$. Since $Y$ is a Banach space, every Cauchy sequence in $Y$ converges to a limit in $Y$. Thus, $(y_n)$ converges to some $y \in Y$.

How to continue?

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Ok, you got that $Ax_n\to y$. Now write $y=Ax$ for $x\in X$. By the definition of the norm on $X$:

$||x_n-x||=||A(x_n-x)||_Y=||Ax_n-Ax||_Y=||Ax_n-y||_Y\to 0$

And so $x_n\to x$.

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