Abstract
In the previous chapter, we have seen how to associate a norm \(\Vert \cdot \Vert \) with a scalar product \(\langle \cdot ,\cdot \rangle \) on a pre-Hilbert space \(H\).
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- 1.
\(L^p(a,b)=L^p([a,b],m)\) where \(m\) is the Lebesgue measure on \([a,b]\). See footnote 7, p. 87.
- 2.
See footnote 5 at p. 147.
- 3.
Two numbers \(1\le p,p'\le \infty \) are said to be conjugate if \(\frac{1}{p}+\frac{1}{p'}=1\), with the convention \(\frac{1}{\infty }=0\).
- 4.
Compare with Exercise 3.9.
- 5.
That is, \(\varLambda \) maps open sets of \(X\) into open sets of \(Y\).
- 6.
This result dates from 1910 (see ([Ko02, p.67)).
- 7.
See footnote 9 at p. 151.
- 8.
- 9.
See footnote 5 at p. 147.
- 10.
Note that \(|y_k|^{p'-2}y_k=0\) if \(y_k=0\) since \(p'>1\).
- 11.
See footnote 5 at p. 147.
- 12.
We refer to p. 98 for the definition of the space \(\fancyscript{C}_c(\mathbb R^N)\).
- 13.
See, for instance, Proposition 2.19 in [Ko02].
- 14.
That is, \(\lim _{\Vert x\Vert \rightarrow \infty } \varphi (x)=\infty \).
- 15.
See Appendix B.
- 16.
This fact is also a direct consequence of Exercise 6.93: \(A_\alpha \) is weakly closed and so, since \(x_{k_n}\in A_\alpha \) for large \(n\), we have \(x_0\in A_\alpha \).
- 17.
A linear operator \(A:X\rightarrow Y\) is said to be compact if it maps any bounded subset of \(X\) into a relatively compact subset of \(Y\). Clearly, any compact operator is also bounded.
- 18.
See footnote 17.
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Cannarsa, P., D’Aprile, T. (2015). Banach Spaces. In: Introduction to Measure Theory and Functional Analysis. UNITEXT(), vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-17019-0_6
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