1
$\begingroup$

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig:

(First, I'm giving verbatim the statements of Kreyszig. )

Theorem 3.3-1:

Let $X$ be an inner product space and $M \neq \emptyset$ a convex subset which is complete (in the metric induced by the inner product). Then for every given $x \in X$ there exists a unique $y \in M$ such that $$\delta \colon= \inf_{\tilde{y} \in M} \Vert x - \tilde{y} \Vert = \Vert x-y \Vert.$$

Lemma 3.3-2:

In Theorem 3.3-1, let $M$ be a complete subspace $Y$ and $x \in X$ fixed. Then $z = x-y$ is orthogonal to $Y$.

Theorem 3.3-4:

Let $Y$ be any closed subspace of a Hilbert space $H$. Then $$H = Y \oplus Z \ \ \ \ \ \ \ Z = Y^\perp.$$

Now my aim is to condense the above three results into one and make them easier and quicker to present. I feel that Kreyszig has repeated things rather superfluously. Am I right?

Here's how I have summarized the above three results into one:

Let $X$ be an inner product space, let $x \in X$, and let $M$ be a non-empty subset of $X$ such that $M$ is complete in the metric induced by the inner product, that is, the restriction to $M \times M$ of the metric $d$ on $X$ given by $$d(u,v) \colon= \sqrt{ \ \langle u-v, \ u-v \rangle \ } \ \ \ \mbox{ for all } u, v \in X.$$
Then we have the following:

(a) If $M$ is a convex subset, then there exists a unique element $y \in M$ such that $$D(x,M) = \Vert x - y \Vert;$$ that is, $$D(x,M) = \sqrt{ \ \langle x-y,\ x-y \rangle \ }.$$

(b) If $M$ is a vector subspace of $X$, then the unique element $y$ asserted in (a) also satisfies $ (x-y ) \perp M$, that is, $$\langle x-y, \ v \rangle = 0 \ \ \ \mbox{ for all } \ v \in M.$$

And, (c) if $X$ is a Hilbert space (i.e. if $X$ is complete in the metric induced by the inner product), if $M$ is a vector subspace of $X$, and if $M$ is closed with respect to the metric induced by the inner product, then there are unique elements $y \in M$ and $z \in X$ such that $$x = y+z \ \ \ \mbox{ and } \ \ \ \langle z, v \rangle = 0 \ \mbox{ for all } \ v \in M. $$

How good an attempt is it? Is it any improvement?

Any advice please? Should I stick with Kreyszig's plan? Or can I improve the above formulation any further?

$\endgroup$
0
$\begingroup$

Your formulation seems compact to me. But here is one more thing that can be done. This formulation in not general but it surely is compact.

Let $H$ be a Hilbert space and let $M$ be a closed subspace of $H$. Then given any $x\in X$ there exists a unique $y\in M$ such that

(a) $\mathrm{dist}(x,M) = \|x-y\|$.

(b) $x-y\perp M$.

(c) $H = M\oplus M^{\perp}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.