Proving Open Mapping of Canonical Projection in Normed Vector Space

[jgens]

Homework Statement

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

Homework Equations

All vector spaces are over \mathbb{R}.

The Attempt at a Solution

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

 

[AKG]

What does the fact that x\in U with U open tell you? Based on what that gets you, you should be able to think of a candidate for the desired open neighbourhood containing [x] and contained in \phi(U).

 

[jgens]

Well I know that there exists 0 < \varepsilon such that B(x,\varepsilon) \subseteq U. But the problem I then have is that ||\cdot||_1 \leq ||\cdot|| so I am having trouble finding a criterion with ||x-y||_1 < \delta implies ||x-y|| < \varepsilon.

 

[micromass]

We wish to prove that B_1(\varphi(x),\varepsilon)\subseteq \varphi(B(x,\varepsilon)). We can take x=0 if we want.

So take z\in B_1(0,\varepsilon), write out what that means.