reflexive space
A vector space is reflexive when the canonical evaluation map V -> (V*)* is not just injective but an isomorphism — that is, when V equals its own double dual via the natural, no-choices map. Reflexivity says 'taking the dual twice gets you all the way home', cleanly and canonically.
Every finite-dimensional space is reflexive, full stop. The evaluation map is always injective, and in finite dimensions dim (V*)* = dim V* = dim V forces injective to be bijective. So for the spaces of a first course, V and (V*)* are simply the same; you may identify a vector with the act of evaluating functionals on it without a second thought.
The story changes in infinite dimensions, and that is the whole reason the word exists. There the evaluation map still injects V into (V*)*, but it can fail to be onto: the double dual can be strictly larger. Spaces where it is onto are the reflexive ones (in functional analysis, using continuous duals: Hilbert spaces and L^p for 1 < p < infinity are reflexive; the spaces c_0 and L^1 and L-infinity are not).
Why care? Reflexivity is what lets you pass freely between a space and its bidual, and it underlies compactness and existence theorems (weak limits, minimizers) that drive much of analysis and optimization. In finite dimensions you get all of this for free; reflexivity names the property that is automatic here but precious later.
Reflexive means the canonical embedding into the double dual fills it completely.
Finite-dimensional spaces are always reflexive; the concept earns its keep only in infinite dimensions, where the algebraic double dual is enormous and even the continuous bidual can outgrow V. Reflexivity is the clean middle case where nothing is lost.