3. а) When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]
3. а) When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]
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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![3. а)
When is a normed linear space called separable? Show that a normed linear space is
separable if its dual is separable [You should state all the proposition or theorems or
corollaries used for proving the theorem]. Is the converse true? Give justification for
your answer. [Whenever an example is given, you should justify that the example
satisfies the requirements.]
b)
Let X be a Banach space, Y be a normed linear space and - be a subset of B (X,Y). If
* is not uniformly bounded, then there exists a dense subset D of X such that for every
x e D,{F(x) :F e } is not bounded in Y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5687772d-b32b-40f6-87ba-be13f259e06b%2Fb4fd090a-8bfb-4ce8-b0ef-ec3c5536fd95%2Fdblm3eb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. а)
When is a normed linear space called separable? Show that a normed linear space is
separable if its dual is separable [You should state all the proposition or theorems or
corollaries used for proving the theorem]. Is the converse true? Give justification for
your answer. [Whenever an example is given, you should justify that the example
satisfies the requirements.]
b)
Let X be a Banach space, Y be a normed linear space and - be a subset of B (X,Y). If
* is not uniformly bounded, then there exists a dense subset D of X such that for every
x e D,{F(x) :F e } is not bounded in Y.
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