ScalingStacks

Theorem 2.4. Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be Banach spaces over $k$, and $f:V\rightarrow W$ be a $k$-linear map.

1. (1)

   The $k$-linear map $f$ is bounded if and only if its graph in $V\times W$ is closed under the product topology.

2. (2)

   Assume that $f$ is bounded and surjective, then $f$ is an open map. In particular, the quotient norm of $\lVert\mathord{\cdot}\rVert_{V}$ on $W$ is equivalent to $\lVert\mathord{\cdot}\rVert_{W}$.

3. (3)

   Assume that $f$ is bounded and injective, then $f(V)$ is closed in $W$.