Functional Analysis

SURPRISE QUIZ I

MTL 411 FUNCTIONAL ANALYSIS



1. A nonzero subspace of a normed linear space can not be bounded - Why?

2. Let X ̸= {0} be a linear space and for x ∈ X, x 7→ ∥x∥ satisfy the properties:
(i) For x ∈ X, ∥x∥ = 0 ⇒ x = 0.
(ii) ∥x + y∥ ≤ ∥x∥ + ∥y∥ ∀ x, y ∈ X.
(iii) ∥αx∥ = |α|∥x∥ ∀ x ∈ X, α ∈ K.

Deduce the following:
(a) ∥0∥ = 0.
(b) ∥x∥ ≥ 0 for all x ∈ X.
(c) x 7→ ∥x∥ defines a surjective map from X to R+ ∪ {0}.
(d) ∥x − y∥ ≥ |∥x∥ − ∥y∥| ∀ x, y ∈ X.

3. Let X be a normed linear space, a ∈ X, 0 ̸= k ∈ K. Prove that the mappings
Ta : X → X, Ta (x) = x + a ∀ x ∈ X;
Mk : X → X, Mk (x) = kx ∀ x ∈ X
are homeomorphisms.

(Recall: Given two metric spaces (X, dX ) and (Y, dY ), a function f : X → Y is called a
homeomorphism if f is bijective and bicontinuous, i.e., f is one-one onto, and f , f −1 are
continuous)

4. Let Y1 and Y2 be subsets of a normed linear space X. Define
Y1 + Y2 = {x + y : x ∈ Y1 , y ∈ Y2 }.
Prove that
(i) If Y1 and Y2 are compact, then Y1 + Y2 is compact.
(ii) If Y1 is compact and Y2 is closed, then Y1 + Y2 is closed.

5. (Bonus Question) If Y1 or Y2 is an open subset of a normed linear space X, prove that
Y1 + Y2 = {x + y : x ∈ Y1 , y ∈ Y2 } is open.




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