Question

Prove that every linear functional on a finite-dimensional topological linear space is automatically continuous.

Short answer

Every linear functional on a finite-dimensional topological space is continuous because it is bounded, which implies continuity.

Step-by-step solution

Define a Linear Functional

Let \( f: V \rightarrow \mathbb{R} \) be a linear functional on a finite-dimensional topological linear space \( V \). This means \( f \) is a linear map, i.e., for any \( u, v \in V \) and \( c \in \mathbb{R} \), we have \( f(u+v) = f(u) + f(v) \) and \( f(cu) = cf(u) \).

Consider the Basis of the Vector Space

Since \( V \) is finite-dimensional, let's say it has dimension \( n \). Choose a basis \( \{v_1, v_2, \ldots, v_n\} \) for \( V \). Any vector \( x \in V \) can be expressed uniquely as \( x = a_1v_1 + a_2v_2 + \ldots + a_nv_n \) where \( a_i \in \mathbb{R} \).

Express the Functional in Terms of the Basis

The linear functional \( f \) can be expressed in terms of the basis as: \( f(x) = f(a_1v_1 + a_2v_2 + \ldots + a_nv_n) = a_1f(v_1) + a_2f(v_2) + \ldots + a_nf(v_n) \). Each \( f(v_i) \) is a constant since \( f \) is a linear functional.

Prove Boundedness Hence Continuity

To show continuity, it's sufficient to show that \( f \) is bounded. Given any vector \( x \in V \), we have \( \|x\| = |a_1|\|v_1\| + |a_2|\|v_2\| + \ldots + |a_n|\|v_n\| \). Now, \(|f(x)| = |a_1f(v_1) + a_2f(v_2) + \ldots + a_nf(v_n)| \leq |a_1||f(v_1)| + |a_2||f(v_2)| + \ldots + |a_n||f(v_n)| \leq M (|a_1| + |a_2| + \ldots + |a_n|) \),where \( M = \max(|f(v_1)|,|f(v_2)|,...,|f(v_n)|) \). Since linearity implies this limiting behavior, \( f \) is continuous.

Conclude Continuity

Since a linear functional on a finite-dimensional vector space is expressible as a finite sum of multiples of its basis elements, and since we can bound these multiples as shown, \( f \) is thus bounded on every ball around the origin, and therefore continuous by definition.

Key concepts

Finite-Dimensional Topological Linear Space

In the world of vector spaces, a finite-dimensional topological linear space is a fascinating concept. Let's break down what it means.

Firstly, "finite-dimensional" means that the space has a limited number of directions or basis vectors. It's like living in a world with a set amount of axes you can move along. For example, a two-dimensional plane allows movement horizontally and vertically, making it two-dimensional.
When we add the "topological" aspect, it implies that the space also has a geometric structure. It isn't just about direction, but also about how these directions are laid out in a continuous manner. A topological space allows us to discuss properties like closeness and continuity.

Ultimately, a finite-dimensional topological linear space combines these ideas, having a finite number of basis vectors with geometric and continuous properties. This blend of linear algebra and topology plays a big role in understanding how functions behave in these structured environments.

Continuity of Linear Functionals

Continuity in mathematics is about smoothness and lack of abrupt changes. Imagine a road without any sharp bends or jumps—this is like a continuous function. For linear functionals, continuity is an important property.

A linear functional is a type of linear map, specifically a function that takes vectors and outputs scalars, often real numbers. When we say it is continuous, we mean that small changes in the input vector lead to small changes in the output.
This avoids "jumpiness" in behavior, making the function predictable and well-behaved.

In practice, every linear functional on a finite-dimensional topological linear space is automatically continuous. This is because in finite dimensions, linearity and boundedness (having a maximum allowable value) ensure continuity. Thus, once you show a linear functional is bounded, continuity follows right behind!

Basis of Vector Space

The concept of a basis in a vector space is crucial for understanding how vectors are formed and manipulated.

Think of a basis as a small set of vectors that can generate or "span" the entire space. In a finite-dimensional vector space, every vector can be uniquely expressed using combinations of these basis vectors.
For example, in a two-dimensional space, a common basis is the pair of axes that define the plane: horizontal and vertical.

To break it down further: if you have a vector space with dimension \(n\), you need \(n\) basis vectors to span the space. Each vector in the space is a unique combination of these basis vectors. This concept is like using letters (basis) to make different words (vectors).
Understanding a basis helps us see how complex vectors are simply orchestrated using a few key components.

Boundedness

Boundedness is a mathematical term that describes a function or set that does not surpass a certain boundary or limit. In simpler words, there is a maximum "size" that something can reach.

For a linear functional, being bounded means that its output remains under a certain maximum value no matter how large the input grows, as long as inputs remain within a specific range. This property ensures that a function doesn't have wild swings in its outputs.
Consider our linear functional \( f \). If we can show that \(|f(x)|\) is always less than or equal to a specific constant times the size of \(x\), we prove boundedness.

• This implies that even as the input grows, the output grows in a controlled manner.
• Boundedness leads to the conclusion of continuity in finite-dimensional spaces.

Hence, when examining linear functionals, proving boundedness is key to demonstrating their continuity.

Linearity

Linearity is a cornerstone of both mathematics and the physical sciences. It refers to functions or operations that preserve addition and multiplication.

In the context of linear functionals, linearity means that two main properties hold:

• Additivity: For any vectors \( u \) and \( v \), \( f(u+v) = f(u) + f(v) \).
• Homogeneity: For any vector \( u \) and scalar \( c \), \( f(cu) = cf(u) \).

These properties ensure that a linear functional treats combinations of vectors in a straightforward, proportional way.

Linearity simplifies problems because it allows complex calculations to be broken down into sums and scalar multiples. When dealing with finite-dimensional spaces, linearity paired with other properties such as boundedness guarantees continuity, showcasing the elegance and predictability of linear structures.