By A.A. Ljapunow, E.A. Stschegolkow and W.J. Arsenin

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**Example text**

Then H is also a subgroup. If H is normal, then so is H . Proof. 16. x/ D gxg 1 we get that g is a / D . 25. C /. Finally, suppose H \ C is closed. Then H is closed. Proof. C /. Suppose p 2 H . Then pW1 1 \ H ¤ ;, so choose q 2 pW1 1 \ H . p 1 q/ 1 2 W1 . pW2 \U / ¤ ;. H \ pW2 / . q 1 H \ q 1 pW2 / since left multiplication by q 1 is a homeomorphism. H \ C / . But H \ C is closed, so q 1 p 2 H \ C H . Hence p D q q 1p 2 H . 23 being used when Y is also a topological group and f W G=H ! Y is a homomorphism.

If B 2 B0 , then 12 B 2 B0 . ˇ/ If B1 , B2 2 B0 , then 9 B3 B0 with B3 B1 \ B2 . on X; making X into a locally convex Then B0 is a base at 0 for a (unique) topologyT space. This space is Hausdorff if, and only if, B0 D f0g. Proof. X; C/ is commutative; (iv)X This is condition (ˇ). 11, the last sentence in this theorem is validated, so it only remains to show that scalar multiplication is jointly continuous. Let F denote the base field, and suppose x0 2 X , c0 2 F, and c0 x0 D y0 2 U , with U open.

PW2 \U / ¤ ;. H \ pW2 / . q 1 H \ q 1 pW2 / since left multiplication by q 1 is a homeomorphism. H \ C / . But H \ C is closed, so q 1 p 2 H \ C H . Hence p D q q 1p 2 H . 23 being used when Y is also a topological group and f W G=H ! Y is a homomorphism. Our final result for this section clarifies just what is required for homomorphisms in general. 26. Suppose G and GQ are topological groups with identity elements e and e, Q respectively. Suppose Be is a neighborhood base at e, and BeQ is a neighborhood base at e.