real analysis – Problems with the Riesz Representation Theorem

I am studying Stanislaw Lojasiewicz book – “An introduction to the Theory of Real Functions” and I do not uderstand few things. I hope you’ll help me. Here is what is written:

G is an open set and $Gamma(G)$ is defined as a class of all continous, non-negative functions on compact space such that $varphileq 1$ and cl${x:varphi(x)neq 0}subset G$,

$lambda(G)=sup_{Gamma(G)}I(varphi).$

Now let $G_n$ be a sequence of open sets. Let $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$. Then there exists $varphiinGamma(bigcuplimits_{i=1}^{infty} G_{i})$ such that $L<I(varphi).$

That’s first thing. Why can we define such constant? How can we know it exist? And why existing of constant $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$ implies fact that $L<I(varphi).$

I will be really thankful for any advices.

Łojasiewicz, Stanisław, An introduction to the theory of real functions. Transl. from the Polish by G. H. Lawden, ed. by A. V. Ferreira, Wiley-Interscience Publication. Chichester (UK) etc.: Wiley. ix, 230 p. textsterling 24.95 (1988). ZBL0653.26001.

real analysis: convergence of Lp of Riesz media for Schwarz functions

by $ delta> 0 $ and $ lambda> 0 $, define the means of Riesz of $ f $ by
$$
S_ lambda ^ delta f (x) = (2 pi) ^ {- n} int _ { mathbb {R} ^ n} e ^ {i langle x, xi rangle} (1 – | xi / lambda |) _ + ^ delta hat {f} ( xi) d xi,
$$
where $ t _ + ^ delta = t ^ delta $ for $ t> 0 $ and zero otherwise. Define the critical index for $ L ^ p ( mathbb R ^ n) $
$$
delta (p) = max {n | frac {1} {2} – frac {1} {p} | – frac {1} {2}, 0 }.
$$

Assume that (1) $ n geq 3 $ and $ p in (1, frac {2 (n + 1)} {n + 3}) cup ( frac {2 (n + 1)} {n-1}, infty), $ or
(two) $ n = $ 2 and $ 1 leq p leq infty. $ Suppose we know that $$ || S_ lambda ^ delta f || _ {L ^ p ( mathbb R ^ n)} leq C_ {p, delta} || f || _ {L ^ p ( mathbb R ^ n)} $$ for all $ lambda> 0 $, where $ f en L ^ p ( mathbb R ^ n) $ and $ delta> delta (p). $

Leave $ g in mathcal {S} ( mathbb R ^ n), $ that is, a function of Schwarz. Show that $ S_ lambda ^ delta g to g $ in $ L ^ p $.

Remark: keep in mind that $ S_ lambda ^ delta g to g $ pointed everywhere, by the Fourier investment formula. I tried to show this motto, but it turns out to be false.

Context: I am studying $ S $ 2.3 Riesz means in $ mathbb R ^ n $ in Fourier integrals in classical analysis, 2nd edition by Sogge. The above statement is necessary in the proof of Corollary 2.3.2.

linear algebra – Existence of adjunct through the Riesz representation theorem

In Well Done Linear Algebra we have a theorem that establishes

Riesz representation theorem: Suppose $ V $ it's finite-dimensional and $ A $ is a linear functional in $ V $. Then there is a unique vector $ u $ such that for each $ v $ : $ A (v) = langle u, v rangle $

However, then the existence of the attached transformation is cited using this theorem

$ langle T v, w rangle = left langle v, T ^ {*} w right rangle $

To see why the above definition makes sense, let's suppose $ T in
mathcal {L} (V, W) $
. Pin up $ w in W $. Consider the linear functionality V
what maps $ v in V text {a} langle T v, w rangle $; this linear
functional depends on $ T $ Y $ w $. By the Riesz representation theorem,
there is a unique vector in V such that this linear functional is
given taking the inner product with him. We call this unique vector.
$ T ^ {*} w $

I do not understand what the two situations are like. First we did not have a transformation before the internal product and now we do. How does the guarantee continue to exist? Not only that, but $ v $ Y $ w $ It could belong to different dimension spaces and $ T $ transforms $ V $ to $ W $. How is the Riesz representation theorem well supported in this case? The two seem quite disconnected.