- #1

jgens

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## Homework Statement

Let [itex](X,||\cdot||)[/itex] be a normed vector space and suppose that [itex]Y[/itex] is a closed vector subspace of [itex]X[/itex]. Show that the map [itex]||x||_1=\inf_{y \in Y}||x-y||[/itex] defines a pseudonorm on [itex]X[/itex]. Let [itex](X/Y,||\cdot||_1)[/itex] denote the normed vector space induced by [itex]||\cdot||_1[/itex] and prove that the canonical projection [itex]\phi:X \rightarrow X/Y[/itex] is an open mapping.

## Homework Equations

All vector spaces are over [itex]\mathbb{R}[/itex].

## The Attempt at a Solution

So far I have been able to show that [itex]||\cdot||_1[/itex] is a pseudonorm, but I am having difficulty showing the canonical projection is an open mapping. Obviously we need to take [itex]U[/itex] open in [itex]X[/itex] and then take [itex][x] \in \phi(U)[/itex]. From here we need to construct an open neighborhood around [itex][x][/itex] which is contained in [itex]\phi(U)[/itex] but I am having difficulty doing this. Any help?