# Zbigniew Semadini's Banach Spaces of Continuous Functions PDF

By Zbigniew Semadini

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**Extra info for Banach Spaces of Continuous Functions**

**Example text**

3. If L is any lattice, then a subgroup LO Ä L is full in L if LO is the maximal subgroup of L that is contained in the span of LO as a vector space. 1. Let H be a standard point group acting on the lattice L; suppose . 1/ 2 H . We let H C denote the orientation-preserving subgroup of H . 1. LC ; H /. 2. L0 ; H /, where LC Ä L0 and each of the subgroups hx; yi, hzi is full in L0 . C 3. L; H / the subgroups hxi, hyi, hzi is full in L0 . 4. L0 ; H /, where LP Ä L0 and each of the subgroups hv2 ; v3 i, hv1 i is full in L0 .

L; H / 38 K D H . ) Now note that H 1 is a subgroup of hH; . 1/i satisfying (i)–(iv). It follows that H 1 D H . L0 ; H /, where L0 is one of the lattices that is paired with hH; . 1. We should next show that there are no repetitions in our list, but the proof of the latter fact follows the pattern from the final paragraph of the proof of (1). 3. Suppose that H D D40 . L; H /. Add the element . L; hH; . 1/i/. L; hH; . L0 ; hH; . x C y C z/ : 2 Let us suppose that L D L0 and hH; . 1/i 1 D hH; . 1/i.

3/, h ¤ 1, and ` is the unique line fixed by h, then P D fv 2 R3 j v ? `g contains a non-zero element of L. 3. If h 2 H is reflection in the plane P , then L \ P is free abelian of rank two. S. J. L; H / Proof. 1. Let v 2 L be such that v 6? ` and v … `. Thus, we can write v D v1 C v2 , where v1 ? `, v2 2 `, and neither v1 nor v2 is 0. 1 C h C : : : C hn 1 /v1 ; since v2 is fixed by h. 1 C h C h2 C : : : C hn 1 /v1 is h-invariant. 1 C h C h2 C : : : C hn 1 /v1 2 `, and therefore can only be 0 (since it is perpendicular to v2 2 ` f0g).