Question

1. Decide whether each of the following is an inner product space. Justify your answers. (i) V = Mnxn(R) with (A, B) = tr(AB). (ii) V = M2x2(C) with (A, B) = tr (iii) V = P(R) with (f,g) = f(1)g(1). (iv) V = P(R) with :((1 ;-) B-4). (v) V is the collection of continuous functions from (0, 1) to C, and (5.9) = 'rg() dt. 4.s)-(sat).

Answer 1

< > 2 # Let x, y, z € V a vector space over F. Then a map i VXV f is said to be inney product if it_satisfies following properties: (x, y, z) a) <xt2, yz <x,y>+<z, y> b) <cx, y> C <xyz c) 2x4) <y, as d) <z>> ze x 0. (i) V = Mnen (R) with <A,B> = tor (AB) take A = tor (A2) =0 But A to > <,> is not imer product (2) V= M2x2(C) with the <ABBR - til 1940.) BMA Here we need to berove (©) property, Rest_ause con be proved using Matrix tot ce :) DA for (AT) tela Consider <A, B> test propertifs =tr = for BTA =tor () of 3) 06) - to ( 14 +1 (1 tor( (!)(** area ON ased = tey tor བ 'll LAB CS Scanned with CamScanner

V= PCB) = { 2, +42+@_22+ <fig> fog (1) ten zal aj ER} let f = X-) then '<fif>= f(b) f0) = beet FED tento aj GR So <,> So <> is not an inner product fy) ve PCR) = {acta, (ao ta, ata x² <+18>= {f}(+) 3'8 di take, f = 2 then <fiff>=0 But f&o is not an inner product 15 V = {f:[0,1) ► ¢ f is continuous <fig> f(t) get at we'll prove pesuberty d). Rest are simple to be driven. <fif>= 5f(+)dt ifff30 Because f is continuoly. So, by property of Riemann integral If iff fo an inner product WA = 0 ejea "ON aбed CS Scanned with CamScanner