Question

Show that all closed hyperplanes of \(C[0,1]\) are isomorphic to \(C[0,1]\) and \(C[0,1] \oplus \mathbf{R}\) is isomorphic to \(C[0,1]\). Consider the subspace \((C[0,1])_{0}\) formed by all functions in \(C[0,1]\) that vanish at 0 . Use it to show directly that \(C[0,1] \oplus \mathbf{R}\) is isomorphic to \(C[0,1]\).

Short answer

All closed hyperplanes of \(C[0,1]\) are isomorphic to \(C[0,1]\). The space \(C[0,1] \oplus \, \mathbf{R}\) is isomorphic to \(C[0,1]\).

Step-by-step solution

Understand the Hyperplane Concept

A hyperplane in a Banach space is a closed subspace of codimension 1, meaning it can be expressed as the kernel of a continuous linear functional. In the context of the Banach space \(C[0,1]\), any closed hyperplane can therefore be seen as a subspace where a certain linear condition is satisfied.

Define the Hyperplane

Consider the subspace \((C[0,1])_{0}\), which consists of all functions in \(C[0,1]\) that vanish at 0: \[(C[0,1])_{0} = \{ f \in C[0,1] \, \mid \, f(0) = 0 \} \]

Identify the Isomorphism for Hyperplanes

Demonstrate that \((C[0,1])_{0}\) is isomorphic to \(C[0,1]\). Consider the map that shifts the argument of functions: \[T(f)(x) = f(x) - f(0)\]This transformation satisfies:\[T(f(0)) = f(0) - f(0) = 0\]Thus, \(T\) maps functions in \(C[0,1]\) to functions that vanish at 0.

Construct Back-Transformation

Next, define an inverse transformation \(T^{-1}\) by:\[T^{-1}(g)(x) = g(x) + \text{constant}\]This will map every function that vanishes at 0 back to its original form. Thus, \(T\) and \(T^{-1}\) provide the required isomorphisms.

Establish the Second Isomorphism

For the second part, consider the direct sum \(C[0,1] \oplus \, \mathbf{R}\), which essentially adds a scalar to each function. Define a bijective linear map \(S\) such that:\[S((f,\text{c}))(x) = f(x) + \text{c}\]This map is clearly bijective and preserves the structure, proving isomorphism.

Verify the Properties

Verify that all linearity and bijectiveness conditions hold for the map \(S\) constructed in Step 5. For any two elements \((f_1, c_1), (f_2, c_2) \) we have:\[S((f_1 + f_2, c_1 + c_2)) = f_1 + f_2 + c_1 + c_2 = S((f_1, c_1)) + S((f_2, c_2))\]And for any scalar \(a\):\[S(a(f, c)) = S((af, ac)) = af + ac = a(f + c) = aS((f, c))\]Thus confirming the isomorphism.

Key concepts

Hyperplanes in Banach Spaces

In Functional Analysis, a hyperplane in a Banach space is a closed subspace with codimension 1. This means it can be described using the kernel of a continuous linear functional. For example, in the Banach space of continuous functions over \([0,1]\), denoted as \(C[0,1]\), a hyperplane is a subspace where a specific linear constraint is satisfied. Consider the subspace \((C[0,1])_{0}\), which includes all functions in \(C[0,1]\) that are zero at \(0\):
\[(C[0,1])_{0} = \{ f \in C[0,1] \mid f(0) = 0 \}\]
This set forms a closed hyperplane in \(C[0,1]\), as it defines a linear condition where functions must vanish at the origin.

Isomorphisms in Banach Spaces

An isomorphism between Banach spaces is a bijective linear map that preserves and reflects the space’s structure. Let's demonstrate an isomorphism between \(C[0,1]\) and \((C[0,1])_{0}\). Define a transformation \(T: C[0,1] \to (C[0,1])_{0}\) by shifting each function by its value at 0:
\[T(f)(x) = f(x) - f(0)\]
This ensures \(T(f)(0) = f(0) - f(0) = 0\), mapping functions in \(C[0,1]\) to those vanishing at 0. The inverse transformation is defined by:
\[T^{-1}(g)(x) = g(x) + g(0)\]
This inverse mapping verifies that \(T\) is indeed bijective and linear, establishing the isomorphism.

Subspaces in Functional Analysis

Subspaces are integral elements within functional analysis, representing subsets that retain the main properties of the parent space. Consider \((C[0,1])_{0}\), a subspace of \(C[0,1]\) made up of functions that vanish at 0. This subspace is important because it helps illustrate the structure and properties of other sets and spaces within analysis. Recognizing the relation between \((C[0,1])_{0}\) and \(C[0,1]\) demonstrates how subspaces can model complex spaces and transformations more manageably. Subspaces like \((C[0,1])_{0}\) are useful for creating mappings and exploring relationship structure within and across Banach spaces.

Direct Sum of Spaces

The direct sum of spaces, denoted as \(C[0,1] \oplus \mathbf{R}\), combines two spaces while preserving their individual structures. For instance, each element in \(C[0,1] \oplus \mathbf{R}\) is a pair \((f, c)\), where \(f \in C[0,1]\) and \(c \in \mathbf{R}\). Establishing that \(C[0,1] \oplus \mathbf{R}\) is isomorphic to \(C[0,1]\) involves constructing a linear map \(S((f, c))(x) = f(x) + c\). By maintaining linearity and bijection, \(S\) shows that each function-space pair can seamlessly transform into \(C[0,1]\). This demonstrates the cohesive and additive structure of direct sums in functional analysis, useful for understanding complex space interactions.