Isomorphic embedding ($c\Vert x\Vert \le \Vert Tx\Vert$)

An isomorphic embedding of normed vector spaces is an injective bounded linear operator whose inverse on its image is bounded. Equivalently $\Vert Tx\Vert \ge c\Vert x\Vert$ for some $c>0$. The two-sided bounds make the domain and its image carry equivalent norms. If the domain is a Banach space, a convergent sequence $T{x}_n$ has a Cauchy sequence of preimages, so its limit remains in the image; thus the image is closed.