diff --git a/stdlib/LinearAlgebra/src/rowvector.jl b/stdlib/LinearAlgebra/src/rowvector.jl
index 1b63a35d70625..493e19b8f46d0 100644
--- a/stdlib/LinearAlgebra/src/rowvector.jl
+++ b/stdlib/LinearAlgebra/src/rowvector.jl
@@ -60,3 +60,7 @@ struct RowVector{T,V<:AbstractVector} <: AbstractMatrix{T}
     end
 end
 const ConjRowVector{T,CV<:ConjVector} = RowVector{T,CV}
+
+# Issue #24590: an inner product on a vector space induces an inner product on the dual space.
+dot(x::Transpose{<:AbstractVector{<:Number}}, y::Transpose{<:AbstractVector{<:Number}}) = dot(transpose(x), transpose(y))
+dot(x::Adjoint{<:AbstractVector{<:Number}}, y::Adjoint{<:AbstractVector{<:Number}}) = dot(adjoint(y), adjoint(x)) # note order (y,x): this is to get the correct conjugation
\ No newline at end of file
diff --git a/stdlib/LinearAlgebra/test/adjtrans.jl b/stdlib/LinearAlgebra/test/adjtrans.jl
index 9dd5b068a5dcd..5fc73ac47e576 100644
--- a/stdlib/LinearAlgebra/test/adjtrans.jl
+++ b/stdlib/LinearAlgebra/test/adjtrans.jl
@@ -460,4 +460,12 @@ end
     @test B == A .* A'
 end
 
+@testset "Issue #24590" begin
+    for T in (Transpose, Adjoint)
+        @test dot(T([1, 2, 3, 4]), T([2, 3, 6, 10])) == 66
+        @test dot(T([1, 2, 3, 4]), (1+2im)*T([2, 3, 6, 10])) == (1+2im)*66
+        @test dot((1+2im)*T([1, 2, 3, 4]), T([2, 3, 6, 10])) == (1-2im)*66
+    end
+end
+
 end # module TestAdjointTranspose