# Theorems (or conjectures) for the theory of proveit.linear_algebra.inner_products¶

In [1]:
import proveit
# Prepare this notebook for defining the theorems of a theory:
%theorems_notebook # Keep this at the top following 'import proveit'.

from proveit import n, a, b, x, y, z, K, H, alpha
from proveit.logic import Forall, InSet, CartExp, InClass, Equals
from proveit.numbers import NaturalPos, Rational, Real, RealNonNeg, Complex
from proveit.numbers import Add, Mult, Abs, LessEq
from proveit.linear_algebra import (VecSpaces, InnerProdSpaces, VecAdd, ScalarMult,
InnerProd, Norm)

In [2]:
%begin theorems

Defining theorems for theory 'proveit.linear_algebra.inner_products'
Subsequent end-of-cell assignments will define theorems
'%end theorems' will finalize the definitions

In [3]:
inner_prod_space_is_vec_space = Forall(
K, Forall(H, InClass(H, VecSpaces(K)),
domain=InnerProdSpaces(K)))

Out[3]:
inner_prod_space_is_vec_space (conjecture without proof):

Number field vector sets are inner product spaces:

In [4]:
rational_vec_set_is_inner_prod_space = Forall(
n, InClass(CartExp(Rational, n), InnerProdSpaces(Rational)),
domain = NaturalPos)

Out[4]:
rational_vec_set_is_inner_prod_space (conjecture without proof):

In [5]:
real_vec_set_is_inner_prod_space = Forall(
n, InClass(CartExp(Real, n), InnerProdSpaces(Real)),
domain = NaturalPos)

Out[5]:
real_vec_set_is_inner_prod_space (conjecture without proof):

In [6]:
complex_vec_set_is_inner_prod_space = Forall(
n, InClass(CartExp(Complex, n), InnerProdSpaces(Complex)),
domain = NaturalPos)

Out[6]:
complex_vec_set_is_inner_prod_space (conjecture without proof):

As special cases, number fields are also inner product spaces:

In [7]:
rational_set_is_inner_prod_space = InClass(Rational,
InnerProdSpaces(Rational))

Out[7]:
rational_set_is_inner_prod_space (conjecture without proof):

In [8]:
real_set_is_inner_prod_space = InClass(Real,
InnerProdSpaces(Real))

Out[8]:
real_set_is_inner_prod_space (conjecture without proof):

In [9]:
complex_set_is_inner_prod_space = InClass(Complex,
InnerProdSpaces(Complex))

Out[9]:
complex_set_is_inner_prod_space (conjecture without proof):

### Inner product linearity properties¶

In [10]:
inner_prod_linearity = Forall(
K, Forall(
H, Forall(
(a, b), Forall(
(x, y, z),
ScalarMult(b, InnerProd(y, z)))),
domain=H),
domain=K),
domain=InnerProdSpaces(K)))

Out[10]:
inner_prod_linearity (conjecture without proof):

In [11]:
inner_prod_scalar_mult_left = Forall(
K, Forall(
H, Forall(
a, Forall(
(x, y),
Equals(InnerProd(ScalarMult(a, x), y),
ScalarMult(a, InnerProd(x, y))),
domain=H),
domain=K),
domain = InnerProdSpaces(K)))

Out[11]:
inner_prod_scalar_mult_left (conjecture without proof):

In [12]:
inner_prod_scalar_mult_right = Forall(
K, Forall(
H, Forall(
a, Forall(
(x, y),
Equals(InnerProd(x, ScalarMult(a, y)),
ScalarMult(a, InnerProd(x, y))),
domain=H),
domain=K),
domain = InnerProdSpaces(K)))

Out[12]:
inner_prod_scalar_mult_right (conjecture without proof):

In [13]:
inner_prod_vec_add_left = Forall(
K, Forall(
H, Forall(
(x, y, z),
domain=H),
domain = InnerProdSpaces(K)))

Out[13]:

In [14]:
inner_prod_vec_add_right = Forall(
K, Forall(
H, Forall(
(x, y, z),
domain=H),
domain = InnerProdSpaces(K)))

Out[14]:

### Our specific Norm definition satisfy the properties of an abstract normalization¶

In [15]:
norm_is_nonneg = Forall(
K, Forall(H, Forall(x, InSet(Norm(x), RealNonNeg),
domain=H),
domain=InnerProdSpaces(H)))

Out[15]:
norm_is_nonneg (conjecture without proof):

In [16]:
scaled_norm = Forall(
K, Forall(H, Forall((alpha, x), Equals(Norm(ScalarMult(alpha, x)),
Mult(Abs(alpha), Norm(x))),
domains=(K, H)),
domain=InnerProdSpaces(K)))

Out[16]:
scaled_norm (conjecture without proof):

In [17]:
norm_triangle_ineq = Forall(
K, Forall(H, Forall((x, y), LessEq(Norm(VecAdd(x, y)),

%end theorems

These theorems may now be imported from the theory package: proveit.linear_algebra.inner_products