 ## solved examples of normed spaces

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Prove that the direct product Q X The fact that the norms do in fact satisfy the triangle inequality is not entirely obvious (usually proved via … �����-����\$�*t斤�0�[�^։%T�--\ǋ%��j �8���_eƝ��qԃqGIl�jm���_�ч� �{�\$��B&���lN-���u����:�"����;UH��'��%�W��BL ��HF�@@��-U��Y�cL�{V��E! Show that the canonical linear operator ˚: X!X?? <> endobj There are many examples of normed spaces, the simplest being RN and KN. is bijective (injective and surjective). 1.3 Examples We give some examples of normed linear spaces. Another name for such a space Xis ‘1. 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. Various Notions of Basis 9 6. 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 will be particularly interested in the inﬁnite-dimensional normed spaces, like the sequence spaces ‘p or function spaces like C(K). Banach Space 2.2-1 Definition. x��\�o�8� �?�w�">E�6�>����6�+���q����M A Banach space is a normed linear space that is complete. Weak Convergence and Eberlein’s Theorem 25 11. 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. Normed and Inner Product Spaces Problem 1. @>�e�d�Ǽ\$�������9[z�W�`S;�!&�n�'��cK�\,��v���|�^x�c�q��}��u�}�A1[�����{�߬���~=e��w~/��zJ�n��]�L. 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? 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): ¥. Normed Space: Examples uÕŒnæ , Š3À °[…˛ • BŁ `¶-%Ûn. Chapter 2 Normed Spaces. A Banach space is a complete normed space ( complete in the metric defined by the norm; Note (1) below ). 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. <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 595.32 841.92] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 4 0 obj 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{. stream Solution. Theorem 3.7 – Examples of Banach spaces 1Every ﬁnite-dimensional vector space X is a Banach space. 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. Dual Spaces 23 10. ��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@ Banach Spaces 2.2 Normed Space. We de ne kxk 1= sup j ja jj. We will introduce certain algebraic structures modelled on natural algebras of operators on Banach spaces. Example 1.13 If 1 p < 1, ‘pis the collection of in nite sequences. <> Problem 2. Normed Linear Spaces: Elementary Properties 5 4. Thus `2 is only inner product space in the `p family of normed spaces. [�R�,M!D���������N�-r^��v�� ��-C�l�����f�e�\ ��0��R�}�\$��;Y����N�����-Wp\$��7��� ����� �`�j�6� This is a normed linear space from a result in Hilbert space Deﬁnition. The Hahn-Banach Extension Theorem 20 9. Prove that any ball in a normed space X is convex. endobj A normed space X is a vector space with a norm defined on it. (g) Let {X i} be an inﬁnite sequence of nontrivial normed linear spaces. k 2 are norms on L n i=1 X i. Three Basic Facts in Functional Analysis 17 8. <>>> Also the important Lebesgue spaces Lp(W,S,m) and the abstract Hilbert spaces that we will study later on will be examples of normed spaces. 2 0 obj �ʶHJ#����pF�y�7�ў�Zo_�ٖ����������oا���v����������/��Ų�.V˗/���������g�p�]}9=ᬄ�8S��2Y�]};=)�W���'�&l�;�����[ ���D8]H�\$�� … %PDF-1.5 In fact ‘1is a Banach space. Deﬁnition – Banach space A Banach space is a normed vector space which is also complete with respect to the metric induced by its norm. The most familiar examples of normed spaces are R nand C . First, we use Zorn’s lemma to prove there is always a basis for any vector space. Let Xbe a normed linear space. In the book’s first proper chapter, we will discuss the fundamental notions and theorems about normed and Banach spaces. %���� Choosing w = 1 yields L2[a,b]. Generally speaking, in functional analysis we study in nite dimensional vector spaces of functions and the linear operators between them by analytic methods. endobj 2The sequence space ℓpis a Banach space … 2. 3 0 obj The ‘tricky’ part in Problem 5.1 is the triangle inequality. (a) We say that ffngn2N converges to f2 X, and write fn! Solution 5.8 (). Bounded Linear Transformations 15 7. This chapter is of preparatory nature. 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. Example. Let B(x0;r) be any ball of radius r > 0 centered at x0 2 X, and x;y 2 B(x0;r). A complete inner product space is called a Hilbert space. 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. 1 0 obj Complete Normed Linear Spaces 6 5. Show further that kxk ≤ kxk 2 ≤ kxk 1 ≤ nkxk. Normed Linear Spaces: Examples 3 3. �ˁ������@�?7 ������X�o�[-IYv� �D{|�����! Abstract. Let Xbe a normed linear space (such as an inner product space), and let ffngn2N be a sequence of elements of X.