geometric interpretation of analytical hahn-banach theorem

Question

I understand this interpretation. But how can I see this in example?

Answer 1

In finite dimensions the picture is quite clear, because we have the inner product.

In $\mathbb{R}^2$, each linear functional is specified by taking the inner product with a vector $x = (x_1,x_2)$. The equation $$ \langle x,y\rangle = 1 $$ specifies a line in $\mathbb{R}^2$ at distance $1/\Vert x\Vert$ from $0$. (This is easy geometry.)

Regarding $\mathbb{R}^2$ as a subspace of $\mathbb{R}^3$, we can extend $x$ to the linear functional $x = (x_1,x_2,0)$, which obviously has the same norm as $(x_1,x_2)$. The equation $$ \langle x, y\rangle = 1 $$ (keeping in mind this is the inner product in $\mathbb{R}^3$ now) specifies a plane in $\mathbb{R}^3$, and clearly it contains the line $L$ in $\mathbb{R}^2 \subset\mathbb{R}^3$ from earlier and is the same distance from the origin.