Show that the sup-norm in C[0, 1] does not come from an inner product by showing...
Show that the sup-norm in C[0, 1] does not come from an inner product by showing that the parallelogram law fails for some f, g. Note that the sup-norm in C[0, 1] is ||f||∞ = {|f(x) : x ∈ [0, 1]}.
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Show that the sup-norm in C[0, 1] does not come from an inner
product by showing...
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a n...
Topology
C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
is some norm on a vector spad then there is some inner product such that 1||...
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
Topology (c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise mu...
Topology
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe de...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
1. Consider the vector space R2 with the norm || - ||p. For p = 1,2,...
1. Consider the vector space R2 with the norm || - ||p. For p = 1,2, 00, draw the unit ball B in the norm || - ||p B x R2x|p < 1} What does B, look like for some other p? Note: p has to be in the range 1 < p < oo for || - ||p to be a norm. Also, only the 2-norm is induced by an inner product on R2. (It is induced by the...
Linear Algebra 2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space ...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be def...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
Topology
C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
Topology
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P
(c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
1. Consider the vector space R2 with the norm || - ||p. For p = 1,2, 00, draw the unit ball B in the norm || - ||p B x R2x|p < 1} What does B, look like for some other p? Note: p has to be in the range 1 < p < oo for || - ||p to be a norm. Also, only the 2-norm is induced by an inner product on R2. (It is induced by the...
Linear Algebra
2) General Inner Products, Length, Distance and Angle a) Determine if (u,v)-3uiv,-u,v, is a dot product b) Show that (u.v)-a+a,h,'2 is a product if a, 20 e)Let A-(41 ..)and B-G ) Use inner product on 4 -2 M (A, B aitai +apb +2a to find the length of A, B, namely ll-41 and 1 d) Find the angle between the two matrices above e) Find the distance between the two above matrices 0) For the functions (x)-1 and...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
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