By Rajendra Bhatia
Those notes are a list of a one semester path on useful research given via the writer to moment yr grasp of statistics scholars on the Indian Statistical Institute, New Delhi. scholars taking this direction have a powerful heritage in actual research, linear algebra, degree conception and chance, and the direction proceeds speedily from the definition of a normed linear area to the spectral theorem for bounded selfadjoint operators in a Hilbert area. The publication is organised as twenty six lectures, every one such as a 90 minute category consultation. this can be worthy to lecturers making plans a path in this subject. ready scholars can learn it all alone.
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Extra resources for Notes on Functional Analysis (Texts and Readings in Mathematics)
The Fourier coefficients of a function f in X are the numbers it an = 1 1 f(t)e-2ntdt. 1) -iit The Fourier series of f is the series 00 a, e 27Lt n=-00 One of the basic questions in the study of such series is whether this series converges at each point tin [-7r, 7r], and if so, is its sum equal to f (t)? An example to show that this is not always the case was constructed by Du Bois-Raymond in 1876. The idea was to construct successively worse functions and take their limit. P. P. it is possible to give a soft proof of the existence of a continuous function whose Fourier series diverges at some point.
Then P1, P2 are continuous linear maps from G(A) into X, Y, respectively. The map Pl is a bijection. So, its inverse Pi 1 is a continuous map from X onto G(A), by the Open Mapping Theorem. Since A = P2Pj 1, A is also continuous. 6. What does this theorem say? Let f be any map from X to Y. To say that f is continuous means that if xn -3 x in X, then the sequence f (x,,,) converges to a limit y in Y and y = f (x). The Closed Graph Theorem says that if f is a linear map between Banach spaces, then to prove its continuity we have to show that if xn -* x in X and f (xn) -* y in Y, then y = f (x).
For which the series > a,,,x,,, converges in X. Define the norm of such a sequence as 7' ajxj11 IIail =sup,, 111: j=1 Show that Y is a Banach space with this norm. The map T(a) = > a,,x,,, is a bounded linear operator from Y onto X. ] Some Applications of the Basic Principles 9. Exercise. The algebraic dimension of any infinite-dimensional Banach space can not be countable. ) 10. Exercise. The algebraic dimension of f,, is c, the cardinality of the continuum. Hints : For each t in (0, 1) let xt = (1, t, t2,..