Metamath Proof Explorer


Theorem riesz1

Description: Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of Halmos p. 31. For part 2, see riesz2 . For the continuous linear functional version, see riesz3i and riesz4 . (Contributed by NM, 25-Apr-2006) (New usage is discouraged.)

Ref Expression
Assertion riesz1 T LinFn norm fn T y x T x = x ih y

Proof

Step Hyp Ref Expression
1 lnfncnbd T LinFn T ContFn norm fn T
2 elin T LinFn ContFn T LinFn T ContFn
3 fveq1 T = if T LinFn ContFn T × 0 T x = if T LinFn ContFn T × 0 x
4 3 eqeq1d T = if T LinFn ContFn T × 0 T x = x ih y if T LinFn ContFn T × 0 x = x ih y
5 4 rexralbidv T = if T LinFn ContFn T × 0 y x T x = x ih y y x if T LinFn ContFn T × 0 x = x ih y
6 inss1 LinFn ContFn LinFn
7 0lnfn × 0 LinFn
8 0cnfn × 0 ContFn
9 elin × 0 LinFn ContFn × 0 LinFn × 0 ContFn
10 7 8 9 mpbir2an × 0 LinFn ContFn
11 10 elimel if T LinFn ContFn T × 0 LinFn ContFn
12 6 11 sselii if T LinFn ContFn T × 0 LinFn
13 inss2 LinFn ContFn ContFn
14 13 11 sselii if T LinFn ContFn T × 0 ContFn
15 12 14 riesz3i y x if T LinFn ContFn T × 0 x = x ih y
16 5 15 dedth T LinFn ContFn y x T x = x ih y
17 2 16 sylbir T LinFn T ContFn y x T x = x ih y
18 17 ex T LinFn T ContFn y x T x = x ih y
19 normcl y norm y
20 19 adantl T LinFn y norm y
21 fveq2 T x = x ih y T x = x ih y
22 21 adantl T LinFn x y T x = x ih y T x = x ih y
23 bcs x y x ih y norm x norm y
24 normcl x norm x
25 recn norm x norm x
26 recn norm y norm y
27 mulcom norm x norm y norm x norm y = norm y norm x
28 25 26 27 syl2an norm x norm y norm x norm y = norm y norm x
29 24 19 28 syl2an x y norm x norm y = norm y norm x
30 23 29 breqtrd x y x ih y norm y norm x
31 30 adantll T LinFn x y x ih y norm y norm x
32 31 adantr T LinFn x y T x = x ih y x ih y norm y norm x
33 22 32 eqbrtrd T LinFn x y T x = x ih y T x norm y norm x
34 33 ex T LinFn x y T x = x ih y T x norm y norm x
35 34 an32s T LinFn y x T x = x ih y T x norm y norm x
36 35 ralimdva T LinFn y x T x = x ih y x T x norm y norm x
37 oveq1 z = norm y z norm x = norm y norm x
38 37 breq2d z = norm y T x z norm x T x norm y norm x
39 38 ralbidv z = norm y x T x z norm x x T x norm y norm x
40 39 rspcev norm y x T x norm y norm x z x T x z norm x
41 20 36 40 syl6an T LinFn y x T x = x ih y z x T x z norm x
42 41 rexlimdva T LinFn y x T x = x ih y z x T x z norm x
43 lnfncon T LinFn T ContFn z x T x z norm x
44 42 43 sylibrd T LinFn y x T x = x ih y T ContFn
45 18 44 impbid T LinFn T ContFn y x T x = x ih y
46 1 45 bitr3d T LinFn norm fn T y x T x = x ih y