If \( \displaystyle x=(x_1,x_2,x_3)\in \mathbb{R}^3 \) and \( \displaystyle \Pi \subseteq \mathbb{R}^3 \) is the plane of equation \( \displaystyle ay_1+by_2+cy_3=\alpha \) , then the point-plane distance from \( \displaystyle x \) to \( \displaystyle \Pi \) is:
\( \displaystyle \text{dist} (x,\Pi )=\frac{|ax_1+bx_2+cx_3-\alpha|}{\sqrt{a^2+b^2+c^2}} \) .
We're to work out the real case, but the theorem holds true also in the general case, i.e. for normed vector spaces over arbitrary fields.
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Few things to remember:
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Let \( \displaystyle (X,||\cdot ||) \) be an infinite-dimensional normed vector space over \( \displaystyle \mathbb{R} \) and let \( \displaystyle (X^*,||\cdot ||_*) \) be its normed topological dual space, i.e. the vector space \( \displaystyle X^* := \{ u: X\to \mathbb{R} :\ u \text{ is linear and bounded} \} \) equipped with the dual norm:
\( \displaystyle ||u||_*:= \sup_{x\in X\setminus \{ \mathfrak{o}\}} \frac{|\langle u ,x\rangle |}{||x||} =\sup_{x\in \mathcal{B}} |\langle u,x\rangle | \) .
Here and in what follows: \( \displaystyle \mathfrak{o} \) is the null vector in \( \displaystyle X \) ; \( \displaystyle \langle \cdot ,\cdot \rangle \) is the duality between \( \displaystyle X^* \) and \( \displaystyle X \) , i.e. \( \displaystyle \langle u , x\rangle := u(x) \) ; and \( \displaystyle \mathcal{B} \) is the open unit ball in \( \displaystyle X \) , i.e. \( \displaystyle \mathcal{B} := \{ x\in X:\ ||x|| < 1 \} \) .
We say that a subset \( \displaystyle S\subseteq X \) is an affine subspace of codimension \( \displaystyle N \) iff there exist exactly \( \displaystyle N \) linearly independent functionals \( \displaystyle u_1,\ldots ,u_N\in X^* \) and \( \displaystyle N \) scalars \( \displaystyle \alpha_1,\ldots ,\alpha_N \in \mathbb{R} \) s.t.:
\( \displaystyle x\in S \Leftrightarrow \begin{cases} \langle u_1 ,x\rangle =\alpha_1 \\ \quad \vdots \\ \langle u_N ,x\rangle =\alpha_N \end{cases} \) .
We refer to the \( \displaystyle N \) equations in the previous formula as the affine equations of \( \displaystyle S \) ; the number \( \displaystyle N \) is usually denoted \( \displaystyle \text{codim} S \) .
We call affine hyperplane each affine subspace \( \displaystyle \Pi \subseteq X \) with \( \displaystyle \text{codim} \Pi =1 \) .
If \( \displaystyle \Pi \) is a hyperplane of equation \( \displaystyle \langle u,x\rangle =\alpha \) , we can also use the suggestive notation \( \displaystyle \Pi_{u,\alpha} \) .
Finally we remember that the distance of a point \( \displaystyle x\in X \) from a nonempty subset \( \displaystyle Y\subseteq X \) is the non negative number defined by:
\( \displaystyle \text{dist} (x,Y):=\inf_{y\in Y} ||y-x|| \) .
\( \displaystyle ||u||_*:= \sup_{x\in X\setminus \{ \mathfrak{o}\}} \frac{|\langle u ,x\rangle |}{||x||} =\sup_{x\in \mathcal{B}} |\langle u,x\rangle | \) .
Here and in what follows: \( \displaystyle \mathfrak{o} \) is the null vector in \( \displaystyle X \) ; \( \displaystyle \langle \cdot ,\cdot \rangle \) is the duality between \( \displaystyle X^* \) and \( \displaystyle X \) , i.e. \( \displaystyle \langle u , x\rangle := u(x) \) ; and \( \displaystyle \mathcal{B} \) is the open unit ball in \( \displaystyle X \) , i.e. \( \displaystyle \mathcal{B} := \{ x\in X:\ ||x|| < 1 \} \) .
We say that a subset \( \displaystyle S\subseteq X \) is an affine subspace of codimension \( \displaystyle N \) iff there exist exactly \( \displaystyle N \) linearly independent functionals \( \displaystyle u_1,\ldots ,u_N\in X^* \) and \( \displaystyle N \) scalars \( \displaystyle \alpha_1,\ldots ,\alpha_N \in \mathbb{R} \) s.t.:
\( \displaystyle x\in S \Leftrightarrow \begin{cases} \langle u_1 ,x\rangle =\alpha_1 \\ \quad \vdots \\ \langle u_N ,x\rangle =\alpha_N \end{cases} \) .
We refer to the \( \displaystyle N \) equations in the previous formula as the affine equations of \( \displaystyle S \) ; the number \( \displaystyle N \) is usually denoted \( \displaystyle \text{codim} S \) .
We call affine hyperplane each affine subspace \( \displaystyle \Pi \subseteq X \) with \( \displaystyle \text{codim} \Pi =1 \) .
If \( \displaystyle \Pi \) is a hyperplane of equation \( \displaystyle \langle u,x\rangle =\alpha \) , we can also use the suggestive notation \( \displaystyle \Pi_{u,\alpha} \) .
Finally we remember that the distance of a point \( \displaystyle x\in X \) from a nonempty subset \( \displaystyle Y\subseteq X \) is the non negative number defined by:
\( \displaystyle \text{dist} (x,Y):=\inf_{y\in Y} ||y-x|| \) .
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Exercise:
Let \( \displaystyle (X,||\cdot ||) \) an infinite-dimensional normed vector space and \( \displaystyle \Pi=\Pi_{u,\alpha} \subseteq X \) an affine hyperplane.
1. Prove that for each \( \displaystyle x\in X \) we have:
(*) \( \displaystyle \text{dist} (x,\Pi) = \frac{|\langle u,x\rangle -\alpha|}{||u||_*} \) .
2. Show that the proof of (*) becomes easier if \( \displaystyle X \) is a Hilbert space.
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Hints: 1. Use the definition of \( \displaystyle ||u||_* \) and homogeneity to determine which values of \( \displaystyle r>0 \) make \( \displaystyle r\mathcal{B} \cap \Pi_{u,||u||_*} =\emptyset \) and which don't; use the translation invariance of the distance function to complete the proof.
2. Use a suitable version of the Orthogonal Projection Theorem.
2. Use a suitable version of the Orthogonal Projection Theorem.