Question

Let A be a continuous linear operator mapping a Banach space \(\mathrm{E}\) onto another Banach space \(E_{1}\). Prove that there is a constant \(\mathrm{a}>0\) such if \(\mathrm{B} \in \mathscr{L}\left(E, E_{1}\right)\) and \(\|A-B\|<\mathrm{a}\), then \(\mathrm{B}\) also maps \(\mathrm{E}\) onto (all of) \(E_{1}\).

Short answer

If \( \| A - B \| < a \) and \( a \) is sufficiently small, then \( B \) is also surjective.

Step-by-step solution

Understand the Problem

We need to show that if a linear operator \( A \) is continuous and maps the Banach space \( E \) onto another Banach space \( E_1 \), then we can choose a constant \( a > 0 \) such that any operator \( B \) that is close to \( A \) (in the operator norm sense) will also map \( E \) onto \( E_1 \).

Apply the Inverse Mapping Theorem

Given that \( A \) is a continuous linear operator mapping \( E \) onto \( E_1 \), the Inverse Mapping Theorem ensures that \( A \) has a continuous inverse. This means \( exists M > 0 \) such that for every \( y \in E_1 \), there is an \( x \in E \) with \( Ax = y \) and \( \| x \| \leq M \| y \| \).

Estimate Needed Continuity Threshold

Since \( A \) is continuous and surjective, we take advantage of the fact that small changes to the operator \( A \) will not disrupt this property, provided \( B \) stays sufficiently close to \( A \). Choose \( a \) small enough so that \( \| A^{-1} \| \| A - B \| < 1 \). This choice of \( a \) ensures that \( B \) remains surjective.

Understand the Role of Norms in Operator Behavior

For linear operators, the operator norm \( \| A - B \| < a \) captures how close the operators are to each other. When \( B \) is within this threshold, \( B \) retains the surjectivity property of \( A \) because the perturbation is too small to break the onto condition.

Conclude with Continuity of B

With \( \| A^{-1} \| \| A - B \| < 1 \), \( B \) being a small perturbation of \( A \) means it's also continuous and surjective. Hence, \( B \) maps \( E \) onto \( E_1 \), completing the proof.

Key concepts

Linear Operators

In the world of functional analysis, a **linear operator** is a function that maps elements from one vector space to another, maintaining the operations of addition and scalar multiplication. This means if you have two elements, say \( x \) and \( y \), and scalars \( c \) and \( d \), a linear operator \( A \) will satisfy:

• \( A(cx + dy) = cA(x) + dA(y) \).

This characteristic is crucial because it preserves the linear structure, allowing us to analyze transformations and solve problems using the operator's properties.
Another significant aspect of linear operators, especially in the context of Banach spaces, is whether they are bounded and continuous. A bounded operator means there exists a constant \( M \) such that for all elements \( x \) in the space, \( \|A(x)\| \leq M\|x\| \). Boundedness is equivalent to continuity for linear operators on Banach spaces.
These ideas are central when analyzing how changes, like perturbations, affect the operator's properties, as is the case with operator \( B \) in our exercise.

Continuous Maps

**Continuous maps** are functions that approach certain values as the input approaches a particular point, without sudden jumps. In the context of Banach spaces and linear operators, a continuous function's relevance is tied to boundedness. If an operator is bounded, it is continuous.
In simpler terms, if you slightly change the input, the output won't change too drastically. This continuity is vital when studying mappings such as those from one Banach space to another because it ensures stability and predictability of the transform.

• A continuous operator means, if you look at a sequence of vectors and their images under this operator, if the sequence of vectors converges, so does the sequence of images.
• For the exercise at hand, continuity ensures that if a linear operator \( A \) is perturbed slightly to \( B \), and the perturbation is within a certain threshold, \( B \) will still behave similarly to \( A \).

Thus, continuity safeguards the surjective property of the operator, ensuring our mappings maintain full coverage from \( E \) to \( E_1 \).

Inverse Mapping Theorem

The **Inverse Mapping Theorem** is a fundamental result in functional analysis, especially in Banach spaces. It states that if a linear operator \( A \) is continuous and bijective from a Banach space \( E \) onto another Banach space \( E_1 \), then the inverse operator \( A^{-1} \) is also continuous.
This theorem is crucial for our problem because it guarantees that if \( A \) maps \( E \) onto \( E_1 \), \( A \) has a continuous inverse. This directly implies stability; small perturbations to \( A \), such as creating \( B \), won't lose the "onto" property if the perturbation is within a specific range.Points to remember:

• The theorem helps us find a constant \( a > 0 \) such that for any \( B \) close enough to \( A \) (in terms of the operator norm), \( B \) also becomes a surjective mapping.
• To ensure this, \( \| A^{-1} \| \| A - B \| < 1 \) must hold true, meaning \( B \) will still map the whole space \( E \) onto \( E_1 \), as \( A \) did.

This theorem links continuity with the operator's behavior, offering assurance that the mapping retains its essential features even after slight changes.