The Hahn-Banach theorem allows us to extend linear functionals defined on a subspace of some vector space $V$ to the entire space. Is it possible to construct an explicit example of this in the finite dimensional case? For example, suppose $V = \mathbb{R}^2$ and $U=\mathbb{R} \subset V$. What would be a simple explicit example of the Hahn-Banach theorem, i.e. what are explicit expressions for

  1. $p:V \to \mathbb{R}$ is a sublinear function
  2. $\varphi: U \to \mathbb{R}$ is a linear functional on the linear subspace $U \subset V$ which is dominated by $p$ on $U$.
  3. The linear extension $\psi:V \to \mathbb{R}$ of $\varphi$ to the whole space such that \begin{align} \psi(x) = \varphi(x) \quad \forall x \in U, \\ \psi(x) = p(x) \quad \forall x \in V. \end{align}

In case of $(\mathbb{R}^2, \|\cdot\|_2)$, things are quite simple.

Let $U \le \mathbb{R}^2$ be a subspace and $\phi : U \to \mathbb{R}$ a linear functional. By the Riesz representation theorem, there exists $a \in U$ such that $\phi(x) = \langle x, a\rangle, \forall x \in U$. Then the linear functional $\psi : \mathbb{R}^2 \to \mathbb{R}$ given by the same formula $\psi(x) = \langle x, a\rangle, \forall x \in \mathbb{R}^2$ is the unique Hahn-Banach extension of $\phi$.

Namely, clearly $\psi$ extends $\phi$ and $\|\phi\| = \|a\|_2 = \|\psi\|$ so $\psi$ is a Hahn-Banach extension of $\phi$.

Let $\zeta : \mathbb{R}^2 \to \mathbb{R}$ be another Hahn-Banach extension of $\phi$. By the Riesz representation theorem, there exists $b \in \mathbb{R}^2$ such that $\zeta(x) = \langle x, b\rangle,\forall x \in \mathbb{R}^2$. Since $\zeta$ extends $\phi$, we have

$$\langle x, a\rangle = \phi(x) = \zeta(x) = \langle x, b\rangle, \forall x \in U \implies \langle x, a - b\rangle = 0, \forall x \in U \implies a - b \perp U$$

Since $b = \underbrace{a}_{\in U} + \underbrace{(b - a)}_{\in U^\perp}$, the Pythagorean theorem gives

$$\|a\|_2^2 + \|b - a\|_2^2 = \|b\|_2^2 = \|\zeta\|^2 = \|\phi\|^2 = \|a\|_2^2 \implies b - a = 0 \implies a = b$$

Therefore $\zeta = \phi$.

This explicit construction is in fact always possible when dealing with a Hilbert space, which $(\mathbb{R}^2, \|\cdot\|_2)$ is an example of.

For an explicit example of the above discussion, consider the subspace $Y = \{(x,2x) \in \mathbb{R}^2 : x \in \mathbb{R}\} \le \mathbb{R}^2$ and the linear functional $\phi :Y \to \mathbb{R}$ given by $\phi(x,y) = x$.

An orthonormal basis for $Y$ is $\left\{\frac1{\sqrt{5}}(1,2)\right\}$ so the orthogonal projection $P_Y$ onto $Y$ is given by

$$P_Y(x,y) = \left\langle (x,y),\frac1{\sqrt{5}}(1,2)\right\rangle \frac1{\sqrt{5}}(1,2) = \left(\frac{x+2y}5, \frac{2x+4y}5\right)$$

Now notice that for all $(x,y) \in Y$ we have

$$\phi(x,y) = x = \langle (x,y), (1,0)\rangle = \langle (x,y), P_Y(1,0)\rangle = \left\langle (x,y), \left(\frac15, \frac25\right)\right\rangle$$

Now the above discussion implies that the unique Hahn-Banach extension of $\phi$ is given by $\psi : \mathbb{R}^2 \to \mathbb{R}$ defined as

$$\psi(x,y) = \left\langle (x,y), \left(\frac15, \frac25\right)\right\rangle = \frac{x}2 + \frac{2y}5, \quad\forall (x,y)\in\mathbb{R}^2$$

  • $\begingroup$ I mean actual explicit examples of $p,\varphi$ and $\psi$ in terms of specific functions. I always understand the abstract theory much better when after first seeing an explicit example. $\endgroup$ – csss Jun 6 '18 at 5:55
  • $\begingroup$ @csss I have added an explicit example, have a look. $\endgroup$ – mechanodroid Jun 6 '18 at 20:02
  • $\begingroup$ Excellent, thanks for the added example, I have a much better insight now! $\endgroup$ – csss Jun 22 '18 at 5:45

You could take $$ p(x) = \| x\|_2, \quad \phi(u) = \frac12 u $$

Then a possible extension for $\psi:\mathbb R^2\to\mathbb R$ would be $$ \psi(x) = \frac12 x_1. $$ Another valid extensions would be $$ \psi(x) = \frac12 x_1 + \frac12 x_2, $$ because this functional is also bounded by $p$.


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