Question

We proved the closed graph theorem using the open mapping theorem. Now prove the open mapping principle using the closed graph theorem. Hint: First, prove it for one-to-one maps using the fact that \(\left\\{\left(y, T^{-1}(y)\right)\right\\}\) is closed. For a general case, note that the quotient map is an open map by the definition of the quotient topology.

Short answer

First prove the principle for one-to-one maps using the closed graph theorem. Then extend it to the general case using the properties of quotient maps and topology.

Step-by-step solution

Understand the problem statement

We need to prove the open mapping principle using the closed graph theorem. Initially, we need to focus on one-to-one (injective) maps.

Prove the theorem for one-to-one maps

Consider a continuous linear operator T: X → Y which is one-to-one. By the hint, we need to utilize the closed graph theorem. Let’s examine the set \(\rightarrow {\ {(y, T^{-1}(y)) \}} \) in the product space Y × X. It will be proven that this set is closed.

Using the closed graph theorem

According to the closed graph theorem, if a linear operator between Banach spaces has a closed graph, then the operator is continuous. Here, \(\rightarrow {\ {(y, T^{-1}(y)) \}} \) is the graph of a continuous inverse operator \(\rightarrow {T^{-1}} \). Therefore, T is an open map.

Extend the proof to general case

For the general case where T: X → Y might not be injective, consider the quotient space X/ker(T) with the quotient map q: X → X/ker(T). Here, \(\rightarrow \bar {T} : X / ker(T) \to Y \) is an injective continuous linear operator, implying it’s an open map as just proven.

Apply the quotient map property

Given that \(\rightarrow \bar{T} \) is open and \(\rightarrow q \) is open by definition of quotient topology, the composition T = \(\rightarrow \bar{T} \) ∘ q is open, thus proving the open mapping principle for the general case.

Key concepts

Open Mapping Theorem

The Open Mapping Theorem is an essential concept in functional analysis. It states that any continuous linear operator between Banach spaces (complete normed vector spaces) maps open sets to open sets. This means if you have a continuous linear function between these spaces, the image of any open set under this function will also be open. This theorem is foundational in understanding how functions behave in infinite-dimensional spaces. The theorem has broad applications and helps in proving the invertibility of operators. In simpler terms, it's a guarantee that such transformations don't 'squash' any open sets into 'closed' regions, preserving a level of 'spread' or 'openness'.

We used the Closed Graph Theorem to prove the Open Mapping Theorem in the exercise above. By considering inverse mappings and utilizing the product space properties, we established the openness of these mappings under certain conditions.

Injective Maps

An injective map, or one-to-one function, is a type of function where each element of the range is mapped from a unique element of the domain. In other words, no two different elements in the domain map to the same element in the range. This property is significant because it ensures that the function is 'reversible', at least on its image.

In the exercise, we initially proved the open mapping principle for injective maps. By utilizing the Closed Graph Theorem, we looked at a set in the product space formed by the graph of the inverse operator. We showed that since the inverse operation's graph is closed, the original operator is both continuous and open. Understanding injective maps helps simplify and approach problems in functional analysis, ensuring mappings retain elements' uniqueness and aiding in proving broader mathematical principles.

Quotient Topology

The quotient topology provides a way to construct new topological spaces by 'gluing together' points in an existing space according to an equivalence relation. When we have a topological space X and an equivalence relation ~, the quotient space X/~ is the set of equivalence classes under ~, equipped with the quotient topology. This new topology is defined such that a set is open in X/~ if and only if its preimage under the quotient map is open in X.

In the exercise, the quotient topology was used to extend the open mapping principle to non-injective maps. By considering the quotient map q from the original space X to the quotient space X/ker(T), we demonstrated how this map is inherently open. The injective continuous linear operator on the quotient space, combined with the open nature of q, allowed us to prove that the original operator T is open. This technique is highly useful in functional analysis, enabling us to manage more complex mappings through simpler, well-understood structures.

Banach Spaces

Banach Spaces are complete normed vector spaces, named after the mathematician Stefan Banach. These spaces are pivotal in functional analysis, providing a structured setting where many important principles and theorems apply. The completeness property means that every Cauchy sequence in the space converges to a limit within the space, ensuring no 'gaps' or 'holes'.

The closed graph theorem, central to our exercise, applies specifically to Banach Spaces. It states that if a linear operator between Banach spaces has a closed graph, then the operator is continuous. This was a crucial step in proving the Open Mapping Theorem. Knowing that we are working within Banach spaces allows mathematicians to leverage powerful tools and theorems, making these spaces a cornerstone in advanced mathematical analysis.