In mathematics, particularly in linear algebra, the notation "V ⊕ W" represents the direct sum of two vector spaces, V and W. This means that every element in the resulting space can be uniquely expressed as the sum of an element from V and an element from W. The two spaces must intersect only at the zero vector, ensuring that their contributions to the sum do not overlap. The direct sum is important because it allows for the combination of different vector spaces while maintaining their individual properties. This concept is widely used in various fields, including functional analysis and representation theory, to simplify complex problems by breaking them down into more manageable parts.