Moreover, in the definition $M=B(a,r)$, one could easily forget that the ball on the right hand side of the equation must be taken with respect to $M$ and not to some larger space, where writing $M\subseteq B(a,r)$ does not allow one to make such a mistake. This coincides with the intuition people want to capture by boundedness, though it is equivalent to other definitions. The definition $M\subseteq B(a,r)$ is a good definition for a metric space or subset thereof being bounded. However, one might note that if you want to define a bounded subset $S\subseteq M$, then you would write $S\subseteq B(a,r)$ rather than $S=B(a,r)$, since the ball would be taking place in $M$ rather than intrinsically $S$. Knowing this, the statement that $M\subseteq B(a,r)$ implies that $B(a,r)=M$ since $\subseteq$ is an antisymmetric relation. It is trivial that we have $B(a,r)\subseteq M$ for any $a$ and $r$. In particular, since a ball is defined as Welland, G.V.: Weighted norm inequalities for fractional integrals. Terasawa, Y.: Outer measures and weak type \((1,1)\) estimates of Hardy–Littlewood maximal operators. Strömberg, J.-O.: Weak type \(L^1\) estimates for maximal functions on noncompact symmetric spaces. Princeton University Press, Princeton (1993) Stein, E.M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Stempak, K.: Examples of metric measure spaces related to modified Hardy–Littlewood maximal operators. Stempak, K.: Modified Hardy–Littlewood maximal operators on nondoubling metric measure spaces. Sihwaningrum, I., Sawano, Y.: Weak and strong type estimates for fractional integral operators on Morrey spaces over metric measure spaces. Sawano, Y., Shimomura, T.: Maximal operator on Orlicz spaces of two variable exponents over unbounded quasi-metric measure spaces. Kaplansky states the following on page 130 of Set theory and metric spaces: 'If the Tietze theorem admitted an easier proof in the metric case, it would have been worth inserting in our account. Sawano, Y.: Theory of Besov spaces, 56 Springer, Singapore. begingroup The question made me wonder whether there is a simpler proof of Tietzes extension theorem (generalizing Urysohn) for the case of metric spaces. Sawano, Y.: Sharp estimates of the modified Hardy–Littlewood maximal operator on the nonhomogeneous space via covering lemmas. Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. 66(4), 1383–1406 (2006)Ĭruz-Uribe, D., Fiorenza, A.: \(L\log L\) results for the maximal operator in variable \(L^\) over nondoubling measure spaces. 23(3), 743–770 (2007)Ĭhen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. 116(1), 5–22 (2015)Īhmida, Y., Chlebicka, I., Gwiazda, P., Youssfi, A.: Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces. 164, 213–259 (2002)Īdamowicz, T., Harjulehto, P., Hästö, P.: Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. In particular, there is no such thing as an absolute value, or even a distinguished point (as is 0 for the reals). For this purpose let Cb(X) ff : f 2 C(X) jf(x)j M 8x 2 X for some Mg: It is readily checked that Cb(X) is a normed space under the sup-norm. In order to turn continuous functions into a normed space, we need to restrict to bounded functions. Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Your intuition is correct to an extent, but recall that metric spaces are a very general concept. In general, in a metric space such as the real line, a continuous function may not be bounded.
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