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Functional Analysis I Homework Assignment 4

Prof. Dr. G. Kutyniok, Dr. F. Philipp Summer Term 2015

Exercise 1: 6 Points Let E be a normed and let ϕ : E → K be a linear functional, ϕ 6= 0. Prove the following statements:

(i) There exists x0 ∈ E with ϕ(x0) = 1.

(ii) E = ker ϕ u span{x0} (direct sum). (iii) ϕ ∈ E∗ (i.e. ϕ is continuous/bounded) if and only if ker ϕ is closed.

(iv) ϕ∈ / E∗ if and only if ker ϕ = E. Hint for (iii): Use (ii) and Lemma 2.6 from the lecture.

Exercise 2: 5 Points Let E and F be normed spaces and let L ⊂ E be a linear subspace. Suppose that T : L → −1 F is bijective. Prove that T is closed if and only if T is closed. Now, let F = E = `2 and set

( ∞ ) 2 X 2 2 L := {(xn) ∈ `2 :(nxn) ∈ ` } = (xn) ∈ `2 : n |xn| < ∞ . n=1

Moreover, define the operator T : L → `2 by T (xn) := (nxn), (xn) ∈ L, and prove the following statements. (i) L is a linear subspace.

(ii) T is bijective.

(iii) T −1 is bounded (what is T −1?).

(iv) T is closed.

(v) L is not closed.

Exercise 3: 4 Points Show that on each infinite-dimensional normed space there exists an unbounded linear functional. Hint: You may use the fact that – by Zorn’s Lemma – for each there exists a so-called Hamel . This is a linearly independent set whose span (i.e. all finite(!) linear combinations) is the entire space. Choose a countable subset of this Hamel basis and normalize each of its vectors (i.e. apply x 7→ x/kxk to the vectors).

Exercise 4: 5 Points Let k : [0, 1] × [0, 1] → K be continuous. For f ∈ C[0, 1] define Z 1 (T f)(x) := k(x, y)f(y) dy, x ∈ [0, 1]. 0 Prove that T is a bounded linear operator from C[0, 1] onto itself, i.e. T ∈ L(C[0, 1]). Show furthermore that Z 1 kT k = sup |k(x, y)| dy ≤ kkk∞. x∈[0,1] 0

Show that T has no eigenvalues λ ∈ K with |λ| > kkk∞. Hint for the norm: For x ∈ [0, 1] consider the functions fε(y) = k(x, y)/(|k(x, y)| + ε).

Additional Exercise: +5 Points Let X := C[0, 1] be endowed with the maximum norm. We define the Volterra intgeral operator K : X → X by Z x (Kf)(x) := f(y) dy, x ∈ [0, 1], f ∈ X. 0 Prove the following statements:

• K ∈ L(X) with kKk = 1.

• The range (i.e., the image) of K is not closed in X.

• The perturbation operator of the identity Id by K, i.e. T = Id + K, is surjective.

See the next page for the deadline of this assignment and a short biography of Stefan Banach. Short Biography of Stefan Banach

Stefan Banach (1892-1945) was a Polish . He is generally considered to have been one of the 20th century’s most important and influential . Banach was one of the founders of modern functional analysis and one of the original members of the Lwow School of Mathematics. His major work was the 1932 book, “Theorie´ des operations´ lineaires”´ (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Krakow, Banach enrolled in the Henryk Sienkiewicz Gymnasium, a secondary school, and worked on mathematics problems with his friend Witold Wilkosz. After graduating in 1910, Banach and Wilkosz moved to Lwow. However, Banach returned to Krakow during , and during this time he met and befriended . After Banach solved mathematical problems which Steinhaus considered difficult, he and Steinhaus pu- blished their first joint work. Along with several other mathematicians, Banach formed a society for mathematicians in 1919. In 1920, after had in 1918 regained indepen- dence, Banach was given an assistantship at . He soon became a professor at the Lwow Polytechnic and a member of the Polish Academy of Learning. Later Banach organized the “Lwow School of Mathematics”. Around 1929 he began wri- ting “Theorie´ des operations´ lineaires”.´ After the outbreak of World War II, in September 1939, Lwow was taken over by the So- viet Union. Banach became a member of the Academy of Sciences of and was the dean of the Department of Mathematics of Physics of Lwow University. In 1941, when Germans took over the city, all institutions of higher education were closed to . As a result, Banach had to earn money as a feeder of lice at ’s Institute for Study of and Virology. While the job carried the risk of becoming infected with typhus, it protected him from being sent to slave labor in Germany and other forms of repression. When the Soviets recaptured Lwow in 1944, Banach reestablished the University. Howe- ver, because the Soviets were removing Poles from annexed formerly Polish territories, Banach prepared to return to Krakow. He died in August 1945 after being diagnosed with lung cancer seven months earlier. Some of the notable mathematical concepts named after Banach include Banach spaces, Banach algebras, the BanachTarski paradox, the HahnBanach theorem, the BanachStein- haus theorem, the Banach-Mazur game, the BanachAlaoglu theorem and the Banach fixed-point theorem.

Please submit your homework in your particular tutorial on May 14 or May 15.