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Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...) T₂(x1, x2. X3,) = (x₁, x₂, X₂, X₂, ...) T3 (x1, x2. X3,) = (x₁, X2, X3, X3, ...)... etc Prove that the sequence (T) is strongly operator convergent

Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...) T₂(x1, x2. X3,) = (x₁, x₂, X₂, X₂, ...) T3 (x1, x2. X3,) = (x₁, X2, X3, X3, ...)... etc Prove that the sequence (T) is strongly operator convergent

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 12EQ
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Q2: Let T₁, T₂, be the following bounded linear operators ¹ → f : T₁(x₁, x₂. x3,) = (x₁, X₁, X₁, X₁, ...) T₂(x1, x2. X3,) = (x₁, X₂, X₂, X₂, ...) T3 (X₁, X2. X3,) = (X₁, X2, X3, X3, ...)... etc Prove that the sequence (Tn) is strongly operator convergent. Also prove that (Tn) is not uniformly operator convergent. 11 Tn (x) - Tm(x) >
Transcribed Image Text:Q2: Let T₁, T₂, be the following bounded linear operators ¹ → f : T₁(x₁, x₂. x3,) = (x₁, X₁, X₁, X₁, ...) T₂(x1, x2. X3,) = (x₁, X₂, X₂, X₂, ...) T3 (X₁, X2. X3,) = (X₁, X2, X3, X3, ...)... etc Prove that the sequence (Tn) is strongly operator convergent. Also prove that (Tn) is not uniformly operator convergent. 11 Tn (x) - Tm(x) >
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