Sage-Tips

Can inner product be infinite?

Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space.

Is an integral an inner product?

Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where “1” is the constant function g(x) = 1 on the interval.

What is a finite dimensional inner product space?

Every basis for a finite-dimensional vector space has the same number of elements. This number is called the dimension of the space. For inner product spaces of dimension n, it is easily established that any set of n nonzero orthogonal vectors is a basis. Let {xn: n = 1, 2, …} be a sequence of vectors in the space.

Do all vector spaces have an inner product?

Even if a (real or complex) vector space admits an inner product (e.g. finite dimensional ones), a vector space need not come with an inner product. An inner product is additional structure and it is often useful and enlightening to see what does and what does not require the additional structure of an inner product.

How do you prove inner product space?

The inner product ( , ) satisfies the following properties: (1) Linearity: (au + bv, w) = a(u, w) + b(v, w). (2) Symmetric Property: (u, v) = (v, u). (3) Positive Definite Property: For any u ∈ V , (u, u) ≥ 0; and (u, u) = 0 if and only if u = 0.

Is inner product always real?

Hint: Any inner product ⟨−|−⟩ on a complex vector space satisfies ⟨λx|y⟩=λ∗⟨x|y⟩ for all λ∈C. You’re right in saying that ⟨x|x⟩ is always real when the field is defined over the real numbers: in general, ⟨x|y⟩=¯⟨y|x⟩, so ⟨x|x⟩=¯⟨x|x⟩, so ⟨x|x⟩ is real. (It’s also always positive.)

Does inner product depend on basis?

Therefore the inner product on V that one gets does depend on the choice of basis, and there is no such thing as a standard inner product on a vector space not equipped with additional structure.

Why is every inner product space a normed space?

If V is an inner product space, then v √〈v, v〉 is a norm on V . Taking square roots yields u + v ≤u + v, since both sides are nonnegative. Thus every inner product space is a normed space, and hence also a metric space.