f, if lim n!1 kf fnk = 0; i.e., 8">0; 9N>0 such that n>N =) kf fnk <": (b) We say that ffngn2N is Cauchy if 8">0; 9N>0 such that m;n>N =) kfm fnk <": Exercise 1.11. Then kx¡x0k < r and ky ¡x0k < r: For every a 2 [0;1] we have kax+(1 ¡a)y ¡x0k = k(x¡x0)a+(1 ¡a)(y ¡x0)k • akx¡x0k+(1 ¡a)ky ¡x0k < ar +(1 ¡a)r = r: So ax+(1 ¡a)y 2 B(x0;r): ¥. Show that the canonical linear operator ˚: X!X?? … @>�e�d�Ǽ\$�������9[z�W�`S;�!&�n�'��cK�\,��v���|�^x�c�q��}��u�}�A1[�����{�߬���~=e��w~/��zJ�n��]�L. Example. Weak Convergence and Eberlein’s Theorem 25 11. Let Xbe a normed linear space. Let Xbe a normed linear space (such as an inner product space), and let ffngn2N be a sequence of elements of X. This chapter is of preparatory nature. �ʶHJ#����pF�y�7�ў�Zo_�ٖ����������oا���v����������/��Ų�.V˗/���������g�p�]}9=ᬄ�8S��2Y�]};=)�W���'�&l�;�����[ ���D8]H�\$�� Hilbert space Deﬁnition. Choosing w = 1 yields L2[a,b]. We will be particularly interested in the inﬁnite-dimensional normed spaces, like the sequence spaces ‘p or function spaces like C(K). [�R�,M!D���������N�-r^��v�� ��-C�l�����f�e�\ ��0��R�}�\$��;Y����N�����-Wp\$��7��� ����� �`�j�6� 2The sequence space ℓpis a Banach space … (a) We say that ffngn2N converges to f2 X, and write fn! A complete inner product space is called a Hilbert space. <> Let B(x0;r) be any ball of radius r > 0 centered at x0 2 X, and x;y 2 B(x0;r). Generally speaking, in functional analysis we study in nite dimensional vector spaces of functions and the linear operators between them by analytic methods. This is a normed linear space from a result in real analysis, because we can identify ‘1with L (N; ), where N is the set of natural numbers and is counting measure, that is, (A) is equal to the number of elements of A. A Banach space is a complete normed space ( complete in the metric defined by the norm; Note (1) below ). �ˁ������@�?7 ������X�o�[-IYv� �D{|�����! Normed Space: Examples uÕŒnæ , Š3À °[…˛ • BŁ `¶-%Ûn. The space of measurable functions on [a,b] with inner product hf, gi = Z b a w(t)f(t)g∗(t)dt, where w(t) > 0, ∀t is some (real) weighting function. k 2 are norms on L n i=1 X i. Prove that any ball in a normed space X is convex. Normed Linear Spaces: Elementary Properties 5 4. Normed Linear Spaces: Examples 3 3. Theorem 3.7 – Examples of Banach spaces 1Every ﬁnite-dimensional vector space X is a Banach space. %PDF-1.5 Thus `2 is only inner product space in the `p family of normed spaces.

Complete Normed Linear Spaces 6 5. Deﬁnition – Banach space A Banach space is a normed vector space which is also complete with respect to the metric induced by its norm. Show further that kxk ≤ kxk 2 ≤ kxk 1 ≤ nkxk. Example 1.12 Let Xbe the collection of in nite sequences x= fa 1;a 2;:::g with each a i2C and sup i ja ij<1. 1 0 obj Solution 5.8 (). <>>> ��f������������as?o\$~:���f�^,���/Q�QV���+,�i����~*��o_�b�zz������[:�!�h뿶l��/���_�� +��]Þ&���~�&z�"�ݕE)��-��yJ�?����,���(h�%�(U\$1�x!��)ܞ���z�'��r�ſ�S;Y����n/t3�Z[�e/`��]�`g��.��JO�~��d�­�ӷ�n��7�[]Tɮ鵮�:^��Z�撵^��TMf� A&A'.�`@r9u@ is a normed space with the norm kak p= 0 @ X1 j=1 ja jjp 1 A 1 p: This means writing out the proof that this is a linear space and that the three conditions required of a norm hold. The vector space B( X, Y ) of all bounded linear operators from a normed space X into a normed space Y is it self a normed space with norm defined by || T || = sup{. First, we use Zorn’s lemma to prove there is always a basis for any vector space. <> endobj �����-����\$�*t斤�0�[�^։%T�--\ǋ%��j �8���_eƝ��qԃqGIl�jm���_�ч� �{�\$��B&���lN-���u����:�"����;UH��'��%�W��BL ��HF�@@��-U��Y�cL�{V��E! The fact that the norms do in fact satisfy the triangle inequality is not entirely obvious (usually proved via … NORMED SPACE: EXAMPLES 1.1 Vector Spaces of Functions Recall that a vector space is over a eld F. Throughout this book it is always assumed this eld is either the real eld R or the complex eld C. In the following F stands for R or C. We de ne kxk 1= sup j ja jj. Banach Space 2.2-1 Definition. Suppose you knew { meaning I tell you { that for each N 0 @ XN j=1 ja jjp 1 A 1 p is a norm on CN would that help? In fact ‘1is a Banach space. stream The ‘tricky’ part in Problem 5.1 is the triangle inequality. <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 595.32 841.92] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> x��\�o�8� �?�w�">E�6�>����6�+���q����M 2. %���� There are many examples of normed spaces, the simplest being RN and KN. The most familiar examples of normed spaces are R nand C . 1.3 Examples We give some examples of normed linear spaces. We will introduce certain algebraic structures modelled on natural algebras of operators on Banach spaces. Problem 2. In the book’s first proper chapter, we will discuss the fundamental notions and theorems about normed and Banach spaces. Example 1.13 If 1 p < 1, ‘pis the collection of in nite sequences. This is a normed linear space from a result in 0 Examples of linear spaces 1 1 Metric spaces and normed spaces 1 2 Banach spaces 2 3 Hilbert spaces 5 4 Operator theory 7 5 Operator algebras 9 0 Examples of linear spaces 0.1 Let Xbe a nite dimensional linear space. 2 0 obj 3 0 obj Chapter 2 Normed Spaces. 4 0 obj Abstract. Three Basic Facts in Functional Analysis 17 8.