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Show that T is continuous

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Analysis IV Exercise 13 10. 5. 2004

1. IfT :CR[0,1]R is the linear transformation dened by

T(f) = Z 1

0

f(x)dx

show thatT is continuous.

2. Leth∈L[0,1].

(a) If f is in L2[0,1], show that f h∈L2[0,1].

(b) Let T :L2[0,1]→L2[0,1]be the linear transformation dened by T(f) =hf. Show that T is continuous.

3. LetHbe a complex Hilbert space and lety∈ H. Show that the linear transformation f :H → Cdened by

f(x) =< x, y >

is continuous.

4. (a) If(x1, x2, x3, x4,· · ·)∈l2, show that

(0,4x1, x2,4x3, x4,· · ·)∈l2.

(b) Let T :l2 →l2 be the linear transformation dened by

T(x1, x2, x3, x4,· · ·) = (0,4x1, x2,4x3, x4,· · ·).

Show that T is continuous.

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