4.5.0 BANACH AND HILBERT SPACES

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2 . Normed Spce Norm is mp :V sch tht for ll,v V nd c. 0 0 if nd only if 0. c c 3. + v + v (tringle ineqlity) Exmple: in spce C[, ] of ll continos fnctions in [, ] norm cn e defined s f mx f ( x) C x [,]. Metric Spce Vector spce V is metric spce if there exists fnction ρ :V V sch tht for ll,v,w V. ρ (,) 0 ρ (,v) > 0 for v. ρ(,v) ρ( v,) 3. for ll ρ(,v) ρ(,w) ρ( w,v) (symmetry) + (tringle ineqlity) ρ (,v) is clled the distnce etween,v V. Vector spce with introdced metric is clled metric spce. In the normed vector spce the metric cn e introdced s ρ,v v 3. Inner Prodct Inner prodct is mp (,:V ) V sch tht for ll,v,w V. (,v) ( v,) ( for complex (,v) ( v,). ( α + βv,w) α(,w) + β( v,w) α, β R 3. (,) 0 (,) 0 if nd only if 0 ) Vector spce with introdced inner prodct is clled n inner prodct spce. In inner prodct spce the norm cn e defined s (,) for ll V 4. Convergence Let V e normed (metric spce) nd let f,f V,,,... The seqence f,f,... converges to f if ρ f,f 0 s lim f f 0 f The seqence f V is clled the Cchy seqence (convergent in itself) if ρ f,f 0 s nd m m

3 The vector spce V is clled complete if ll its Cchy seqences re convergent in V. A complete normed spce is clled the Bnch spce. For exmple, n is Bnch spce with x x + x. n A complete inner prodct spce is clled Hilert spce. 5. Orthogonlity In the inner prodct spce,v V re clled orthogonl if,v 0. If set { } V consists of mtlly orthogonl vectors, (,m) 0 when m, then this set is clled n orthogonl set. If in ddition,, then set { } V is clled orthonorml. Orthogonl set is linerly independent set. If set { } V orthonorml set { v } is linerly independent then it cn e converted to the V with the help of the so clled Grm-Schmidt orthogonliztion process: Grm-Schmidt process v v,v v,v v v ( ),v v,v v,v v,v v,v v,v v This lgorithm cn e formlized with the help of Grm s determinnt: G (, ) (, ) (, ) (,) (,) (,) (,) (,) (,), G0 Orthonorml vectors re determined y the forml (, ) (, ) (, ) (, ) (, ) (, ) v GG (, ) (, ) (, ) The orthonorml set { },,... V is sid to e complete if there does not exist vector v 0, v V sch tht it is orthogonl to ll vectors from { }.

4 6. Forier Series Let { } ( f, ) V e n orthonorml set. is clled the Forier series (generlized Forier series) ( f, ) re clled the Forier coefficients, c ( f,) Theorem The Forier series ( f,) fnction f L (,) ( f, ) f is convergent to the if nd only if (Prsevl s eqtion) f x f, Proof: Let Let { } L (,) If for ny f L (,) ( f, ) f ( f,f) ( f,), ( f,) c, c c, ccm,m m c + ( f, ) e n orthonorml set. its Forier series converges to f in L (, ), then { } is sid complete in L,. 7. Vector Spce L Consider prticlr cse of Eqtion 3.3 from Definition 3.3 (p.05), with p I, : nd intervl [ ] L (, ) ϕ : (,) ϕ ( x) < Inner prodct in vector spce L (, ) : For,v L (,) (,v) ( x) v( x) inner prodct in L (, ) define: (,v) ( x) v( x) p( x) weighted inner prodct in L (, ) p with the weight fnction p( x) > 0

5 Inner prodct vector spce L spces., elongs to the clss of Hilert Introdced inner prodct indces the norm in L (, ): ( x) ( x) p( x) p Historiclly, the first complete set ws sed y Forier set of trigonometric fnctions, cos x, sin x π π π the intervl ( 0,π ).,,,... in The complete orthogonl sets sed in the soltion of PDE will e generted y the soltion of the Strm-Lioville prolems Exercizes: The set of monoms {,x,x,x,...} is linerly independent in L,. ) Using the Grm-Schmidt orthogonliztion lgorithm with inner prodct (,v) x v x constrct n orthonorml set in L (,) (the otined set will e the set of the Legendre polynomils p the the sclr mltiple). ) Using the Grm-Schmidt orthogonliztion lgorithm with inner prodct (,v) x v x x constrct n orthonorml set in L (,) (the otined set will e the set of the Tcheyshev polynomils p the the sclr mltiple). c) Use the otined orthonorml sets for generlized Forier series expnsion of the fnction: f ( x) x,0 x 0, Compre the reslts for trncted series with,3,4 terms. Me some oservtions.

6 The Scottish Cfe in Lvov The originl Szoc Cfé (Scottish Cfé) in Amemichn Street in Lvov, Urine (shown t the time when it ws the Dessert Br t Shevcheno Prospet 7). Now it is Bn (on the right). The cfé ws meeting plce for mny mthemticins inclding Bnch, Steinhs, Ulm, Mzr, Kc, Schder, Kczmrz nd others. Prolems were written in oo ept y the lndlord nd often prizes were offered for their soltion. A collection of these prolems ppered lter s the Scottish Boo. R D Mldin, The Scottish Boo, Mthemtics from the Scottish Cfé (98) contins the prolems s well s some soltions nd commentries. Lvov University LVOV