1. Consider the map π1 ∶ FN Ð→ F given byπ1((an)) = a1.(a) Show that π1 is linear
Question
- Consider the map π1 ∶ FN Ð→ F given byπ1((an)) = a1.(a) Show that π1 is linear
Solution
To show that the map π1 : FN → F given by π1((an)) = a1 is linear, we need to show that it satisfies two properties: additivity and homogeneity.
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Additivity: For any two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) in FN, we have:
π1(x + y) = π1((x1 + y1, x2 + y2, ..., xn + yn)) = x1 + y1 = π1(x) + π1(y)
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Homogeneity: For any scalar c in F and any vector x = (x1, x2, ..., xn) in FN, we have:
π1(cx) = π1((cx1, cx2, ..., cxn)) = cx1 = cπ1(x)
Since π1 satisfies both additivity and homogeneity, we can conclude that π1 is a linear map.
Similar Questions
1. Consider the map π1 ∶ FN Ð→ F given byπ1((an)) = a1.(a) Show that π1 is linear.(b) Prove that the restriction of π1 to ℓ∞ or to ℓp for p ⩾ 1 is continuous and surjective
2. Consider the left shift map L ∶ FN Ð→ FN given by L((an)) = (an+1), that isL(a1, a2, a3, . . . ) = (a2, a3, . . . ).(a) Prove that L is a surjective linear map. What is the kernel of L?(b) Prove that for all p ⩾ 1 and for p = ∞, the restriction of L to ℓp is a surjective continuousmap onto ℓp.(c) Define the right shift map R ∶ FN Ð→ FN and prove that it is an injective linear map,the restriction of which is distance-preserving for any ℓp with p ⩾ 1 and p = ∞.(d) Check that L ○ R = idFN ≠ R ○ L
Consider the function g ∶ ℓ2 Ð→ F given byg(x) =∞∑n=1xnn2 .(a) Find y ∈ ℓ2 such thatg(x) = ⟨x, y⟩ for all x ∈ ℓ2.(b) Deduce that g is linear and bounded and find its norm ∥g∥.[Hint: You may use without proof the fact that∞∑n=11n4 = π490 .]
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