# Determine all norm-preserving linear extensions for a functional from the subspace $\{(x,y,z): z=0 \}$ of $\mathbb R^3$.

Given the subspace $$Y=\{(x,y,z): z=0 \}$$ of $$\mathbb R^3$$, I would like to determine all norm-preserving linear extensions for the functional

$$f(x):= a \cdot x$$

where $$a$$ is some fixed vector of the form $$(a_1,a_2,0)$$

Here norm-preserving linear extension $$g$$ is any functional defined on all of $$\mathbb R^3$$, such that $$g(x)=f(x)$$ for $$x \in Y$$, and such that $$\|g \| = \|f \|$$. And such an extension is possible by the Hahn-Banch Theorem.

I'm not sure how to approach the problem but I believe I can see the following:

The function $$f$$ would in fact be defined also as a function from $$\mathbb R^3$$ and would still be linear as such. Thus this is one possible extension?

Is it possible to extend $$f$$ so that it is "an other function on $$\mathbb R^3 \setminus Y$$"? Or will this break down the linearity?

Much grateful for any help provided!

• Using the Euclidean norm? – copper.hat Jan 8 at 20:24
• Yes I'm using the Euclidean norm! – MrFranzén Jan 8 at 20:30

The norm of the functional $$f(x) = a^T x$$ is $$\|a\|$$. On $$Y$$, $$f$$ has norm $$\sqrt{a_1^2 + a_2^2}$$, hence if extended to the entire space, we must have $$\bar{a} = (a_1,a_2,a_3)^T$$ for some $$a_3$$. Hence the only way the norm can be preserved is if $$a_3=0$$.
• Thank you! But how do we make sure that only extensions of the form $g(x) = \bar{a} x$ are possible? – MrFranzén Jan 8 at 20:52
• All linear functionals on $\mathbb{R}^3$ are of the form $b^Tx$ for some $b$. It is immediate that $b_1=a_1$, $b_2=a_2$ by taking $x=(1,0,0)^T$ and $x=(0,1,0)^T$. – copper.hat Jan 8 at 20:54
• Or the fact that $e_1,e_2,e_3$ form a basis for $\mathbb{R}^3$. – copper.hat Jan 8 at 20:59