Let V=C^0(ℝ) be the vector space consisting of all...

Let $V=\mathrm{C}^0(\mathbb{R})$ be the vector space consisting of all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Explain why the set of all functions such that $f(1)=0$ is a subspace, but the set of functions such that $f(0)=1$ is not. For which values of $a, b$ does the set of functions such that $f(a)=b$ form a subspace?

Key Concepts

Vector Space

A vector space is a collection of objects which can be added together and scaled by numbers, and where these operations obey certain axioms. In the context of continuous functions, the set of all continuous functions on ?, denoted C?(?), forms a vector space with the usual function addition and scalar multiplication.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations. To qualify as a subspace, the subset must include the zero vector, and be closed under addition and scalar multiplication.

Subspace Test

The subspace test is a set of criteria used to check if a subset of a vector space is a subspace. Specifically, the subset must contain the zero vector, and for any two elements in the subset, their sum must also be in the subset; moreover, multiplying any element by any scalar must result in another element in the subset.

Zero Element Requirement

An essential condition for a subset to be a subspace is that it must contain the zero element (zero function). For subsets defined by constraints on function values, the condition must allow the zero function (which has value 0 at every point) to satisfy the constraint.

Linear Constraint for Subspace Formation

When considering subsets of a function space defined by a condition of the form f(a) = b, the condition is linear if it is homogeneous, meaning that it equals zero when evaluated at the zero function. Therefore, such a subset forms a subspace if and only if b = 0. If b is nonzero, the zero function does not satisfy the condition, and the set cannot be a subspace.

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