Give an example of a vector space V and subspaces U1, U2 of...

Give an example of a vector space $V$ and subspaces $U_{1}, U_{2}$ of $V$ such that $U_{1} \times U_{2}$ is isomorphic to $U_{1}+U_{2}$ but $U_{1}+U_{2}$ is not a direct sum.
Hint: The vector space $V$ must be infinite-dimensional.

Key Concepts

Vector Space

A vector space is a collection of objects called vectors, where two operations (vector addition and scalar multiplication) are defined and satisfy eight axioms such as associativity, commutativity of addition, existence of an additive identity, and distributive properties. This general algebraic structure is key in many fields of mathematics and physics and can be either finite or infinite dimensional.

Subspace

A subspace is a subset of a vector space that itself is a vector space under the same operations of addition and scalar multiplication. This concept is useful for breaking down complex spaces into simpler pieces and for studying the structure and properties within a larger vector space.

Direct Sum of Subspaces

The direct sum of two subspaces is a way to combine them such that every element of the resulting space can be uniquely written as a sum of one element from each subspace. This requires the subspaces to intersect only at the zero vector. When this uniqueness property fails, the sum is not a direct sum even if it covers the same set of combined elements.

Sum of Subspaces

The sum of two subspaces is the set of all vectors that can be expressed as the sum of a vector from each subspace. This combined space might not have the property of a direct sum if the intersection of the subspaces is more than just the zero vector, leading to non-uniqueness in expressing elements.

Isomorphism

An isomorphism between two vector spaces (or algebraic structures) is a bijective linear map that preserves the vector space structure. Two spaces are isomorphic if they are essentially 'the same' in the context of linear algebra, meaning their structure and operations behave identically under the mapping.

Infinite-Dimensional Vector Spaces

Infinite-dimensional vector spaces are vector spaces that do not have a finite basis. They often appear in functional analysis and other advanced areas of mathematics. These spaces allow for constructions and examples that differ from the finite-dimensional case, providing richer and sometimes counterintuitive structures, such as sums of subspaces that are isomorphic to a product structure even without being direct sums.

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