Example 1.4 (H.W. )(1) Let (V₁, ||- ||1) and (V₂, ||- ||₂2) be two normed spaces over F and let V = V₁ × V₂show that ||. ||: V = V₁ × V₂ → R defined as ||(x, y) || = ||x||₁ + ||y||2 is a norm onV.

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Answered: (1) Let (V₁, ||- ||1₁) and (V₂, ||-…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sketch each of the following sets and decide if they are open sets. Give brief reasons for your decision. (a) The open interval (-1, 1) in the normed vector space (R,| |). (b) The closed interval |-1, 1] in R. (c) The interval [0, 0) in R. (d) The line {(2x, x): a E R} in R². (e) The square (0, 1] x [0, 1) in R2. (f) The set {(0,0, 0)}U (0, 1) x (0, 1) x (0, 1) in R.
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3. Which of the following pairs of vector spaces are isomorphic? Justify your answers. F³ and P3 (F). F4 and P3(F). (a) (b) (c) (d) M2x2 (R) and P3(R). V = {A € M₂x2 (R) : tr(A) = 0} and R4.
Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || x ||x + || y lly and I| (x,y) || . = max (||x||x<||y|ly). Prove that each defines a norm on X x Y and that they are equivalent.
Q4: Let X R³ be a vector space F and||||= ₁² +₂1² +131² V (11, 12, 13) E R³ show that whether (R³, || ||) is normed space or not?
(CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.
Let ||. ||₁ and ||- |l2 be norms on vector space v if ||X→x||₁ → || x₂- Prove that ||. ||₁ and II. ||2 are equivalent x||2
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Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n -vector y E R",3 a set of coefficients {t₁, t₂,... tn} with at least one t; #0 such that 1tixi = y. Prove that the coefficients t₁, t₂,.., t are unique.
Lemma 2.31 Let (X, || · ||x), (Y, || · ||Y) be normed vector spaces and F E L(X,Y). Then ||Fcx,Y) sup xƐX; ||x||x=1 ||Fx||Y sup xƐX; ||x||x<1 ||Fx||y. Proof: excersises.
Prove that the map || · || : L(V, W) → R defined by {I > ||x|| | ||(x)7||} dns = is a norm on L(V, W).
1. Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = V· W. I.e. f preserves the inner product on the vector space R". Show that f is a linear transfor- mation.

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(1) Let (V₁, ||- ||1₁) and (V₂, ||- ||₂2) be two normed spaces over F and let V = V₁ × V₂ show that ||. ||: V = V₁ × V₂ → R defined as ||(x, y) || = ||x||₁ + ||y||2 is a norm on V.