Once you get away from vector spaces of finite dimension, it is no longer true that every linear map is automatically continuous. For example, let Coo {f €F(N, R) f(n) = 0 for all sufficiently large n}, where F(N, R) is the vector space of all func tions from N to R with the usual pointwise operations. Another way of expressing the condition for f to belong to Coo is that {n: f(n) #0} is finite. Then define L: Coo R by L(f) = Enf(n). Notice that the sum on the right hand side is in fact a finite sum. Now (a) Check that L is linear.

Once you get away from vector spaces of finite dimension, it is no longer true that every linear map is automatically continuous. For example, let Coo {f €F(N, R) f(n) = 0 for all sufficiently large n}, where F(N, R) is the vector space of all func tions from N to R with the usual pointwise operations. Another way of expressing the condition for f to belong to Coo is that {n: f(n) #0} is finite. Then define L: Coo R by L(f) = Enf(n). Notice that the sum on the right hand side is in fact a finite sum. Now (a) Check that L is linear.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Which of the following polynomials from the vector space R[x] is linearly expressed by the polynomials from the subset {1+x,1-x^2,x^3)? a.) x + x^2 + x^3 b.) 1+x+x^2 c.) 1+x+x^3 d.) 1 -x^2+x^3 e.) 1+2x+x^2
Let N and N' be normed linear spaces with the some scalars and let T:N-N' be a linear T Transformation then prove that T is bounded if and T only if T is continuous.
Please answer the question in handwritten form
1) Let V, W be F vector spaces. Recall that a skew-symmetric map is a function f: V × V → W such that ƒ(V₁, V2) = −ƒ (V₂, V₁) (a) (b) Prove that an alternating map is always skew-symmetric Prove that if char(F) + 2 then a skew-symmetric map is also alternating
Let X and Y be normed spaces and F:X→Y (a) be a continuous linear map. Show that F is closed . Is the converse true ? Justify.
For any linear map f: EE, let U+ = Ker ((id - f)) and let U¯ = Im(½ (id – ƒ)). If ƒ² = id, then U+ = Ker ((id - )) = m((id + 1)). and so, f(u) = =u on U+ and f(u) = =-u on U-.
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.
Consider two finite dimensional (real) vector spaces V and W with dim(V) = n and dim(W) = k. Define L(V, W) to be the set of linear transformations: L(V, W) = {T: The map T: VW is linear} 1. Define addition and scaling on L(V, W). 2. Sketch a proof that L(V, W) is a vector space. 3. Prove that L(V, W) is finite dimensional by finding an explicit basis for it.
Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤ b is a vector space, with addition and numerical multiplication defined by (f+g)(x) = f(x) + g(x) and (tf)(x) = tf(x).
(CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.
Asap plz
True or false