... Definition N. Let φ(x) = Ax+ b be an isometry of Rn, where A ∈ Rnxn is an orthogonal matrix and b ∈ Rn be an ... isometry of Rn. Define (a) φ is orientation-preserving ⇐⇒ det(A) > 0⇐⇒ det(A) = 1. (b) φ is orientation ... Definition N. Let φ(x) = Ax+ b be an isometry of Rn, where A ∈ Rnxn is an orthogonal matrix and b ∈ Rn be an ... isometry of Rn. Define (a) φ is orientation-preserving ⇐⇒ det(A) > 0⇐⇒ det(A) = 1. (b) φ is orientation ...

FAI HS2022 Problem Set 2

... ) Prove that (c0)∗ ∼= ℓ1, i.e. show that there exists a surjective isometry I : ℓ1 → (c0)∗. (b) Prove that ... (ℓ1)∗ ∼= ℓ∞, i.e. show that there exists a surjective isometry I˜ : ℓ∞ → (ℓ1)∗. (c) Prove that there ... ) Prove that (c0)∗ ∼= ℓ1, i.e. show that there exists a surjective isometry I : ℓ1 → (c0)∗. (b) Prove that ... (ℓ1)∗ ∼= ℓ∞, i.e. show that there exists a surjective isometry I˜ : ℓ∞ → (ℓ1)∗. (c) Prove that there ...